Valuing Individual Player Involvements in Norwegian

Valuing Individual Player Involvements in
Norwegian Association Football
Vegard Rødseth Bjertnes
Olav Nørstebø
Eirik Vabo
Industrial Economics and Technology Management
Submission date: June 2016
Supervisor:
Stein-Erik Fleten, IØT
Co-supervisor:
Magnus Stålhane, IØT
Lars Magnus Hvattum, HiMolde
Norwegian University of Science and Technology
Department of Industrial Economics and Technology Management
Preface
This Master’s Thesis is written as a concluding part for achieving a Master of Science at the
Norwegian University of Science and Technology. The degree specialisation is Investment,
Finance and Financial Management at the Department of Industrial Economics and Technology
Management. Two of the authors have technical background in Mechanical Engineering and
one author in Energy and Environmental Engineering. All authors have specialised in Empirical
and Quantitative Methods in Finance. The study has been conducted over the spring of 2016.
This work started as an initiative from Rosenborg Ballklub, a Norwegian top division association football team. They have been inspired by the focus on statistical analysis in football
in recent years, and wants to shift towards making decisions supported by quantitative methods
in combination with qualitative considerations. This enquiry from Rosenborg was welcomed
at the Department of Industrial Economics and Technology Management and resulted in this
thesis.
Trondheim, 10.06.2016
Vegard Bjertnes
Olav Nørstebø
Eirik Vabo
Summary
Evaluation of player performance in association football has to a large extent been limited to
subjective opinions and simple, easily observable parameters such as goals, passing accuracy
and ball recoveries. In this thesis, three models with the goal of objectively rating players by
looking at individual actions are presented and documented.
An expected goals (xG) model is developed by looking at 13,440 shots attempted in 480
football matches in the Norwegian top division, Tippeligaen. The likelihood of scoring is estimated using binary logistic regression with ten explanatory variables. This model is used as a
foundation to evaluate the performance of players with regard to their shot efficiency.
Two variations of a zero-sum two-agent Markov game model based on matches from two
seasons are developed in order to evaluate other actions than shots. The large state spaces
contain three contextual parameters: time period, match status and manpower difference. In
addition, different field zones are used in the definition of a state. Reinforcement learning
through a Q-function is applied to learn the value of each state and state-action pairs. Players
are rated by their impact per 90 minutes played, and results are presented as top 10 lists of
players in each position.
The reliability of the models is assessed by looking at correlations across seasons. Validity
of the two Markov models are examined through comparisons to two subjective player ratings
and one provider of market value estimates of players. The models are also tested out of sample.
Areas in which extensions of the three models seem possible or appropriate are addressed and
highlighted.
iii
Sammendrag
Evaluering av spillere i fotball har lenge vært begrenset til subjektive meninger og enkle, lett
observerbare parametre som skårede mål, pasningssikkerhet og ballgjenvinninger. Tre modeller
er utviklet og dokumentert i denne masteroppgaven, med mål om å evaluere spillere objektivt.
En expected goals (xG)-modell er utviklet ved å se på 13,440 skuddforsøk fra 480 fotballkamper spilt i Tippeligaen. Sannsynligheten for å skåre på et skudd er estimert ved hjelp av
binær logistisk regresjon med ti forklaringsvariable. Denne modellen er videre brukt til å evaluere spilleres skudd-effektivitet.
To versjoner av en nullsum to-agents Markov game-modell er utviklet basert på kamper fra
to sesonger for å evaluere andre involveringer enn skudd. Et stort sett av tilstander inneholder
tre kontekstvariable: tidsperiode, kampstatus og forskjell i antall spillere på banen. I tillegg fotballbanen delt inn i soner som også en del av en tilstand. Maskinlæringsteknikken forsterkende
læring er brukt gjennom en Q-funksjon til å lære verdien av hver tilstand, og par av tilstander
og handlinger. Spillere er rangert etter hvor stor påvirkning de hadde på kamper per 90 spilte
minutter, og resultatene er presentert som topp 10-lister for hver posision.
Påliteligheten til modellene er vurdert på grunnlag av korrelasjoner på tvers av sesonger.
Validiteten av de to Markov-modellene er vurdert gjennom sammenligning med to subjektive
spillerrangeringer og estimater på markedsverdien til spillerene. Modellene er også testet på
data utenfor utvalget de er lært på. Områder der de tre modellene kan utvides og forbedres er
presentert i slutten av oppgaven.
iv
Acknowledgments
First we want to thank our supervisors Magnus Stålhane, Lars Magnus Hvattum and Stein-Erik
Fleten. Their commitment and interest in our work have motivated us and made the progress
steady and efficient. Without their quick and thorough feedback this thesis would never been a
reality.
We also want to thank Rosenborg Ballklub for providing us the opportunity to work with
statistical analysis and problem solving within an area we all have a passion for. The kind welcome received from Rosenborg Ballklub and the close collaboration with Stig Inge Bjørnebye
made this project a unique experience.
Finally we want to thank Opta Sports, Kenneth Wilsgård from Norsk Toppfotball and The
Planning, Optimization and Decision Support Group at Molde University College for providing
us with the data used in this thesis.
v
vi
Table of Contents
Preface
i
Summary
iii
Sammendrag
iv
Acknowledgments
v
Table of Contents
ix
List of Tables
xiii
List of Figures
xvi
1
2
3
Introduction
1.1 Motivation . . . . .
1.2 Research Questions
1.3 Limitations . . . .
1.4 Report Structure . .
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Literature Review
2.1 Sports Analytics in Football . . . . . . . .
2.2 Expected Goals (xG) . . . . . . . . . . . .
2.3 Markov Models in Sports Analytics . . . .
2.4 Evaluation of Individual Player Performance
2.5 Financial Valuation of Football Players . . .
Basic Theory
3.1 Binary Logistic Regression . . . . .
3.2 Markov Games . . . . . . . . . . .
3.2.1 Definition . . . . . . . . . .
3.2.2 Policies and Value Functions
3.2.3 Value of a Policy . . . . . .
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vii
4
5
Experimental Setup
4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Expected Goals Model . . . . . . . . . . . . . . . . . .
4.2.1 Model Building Approach . . . . . . . . . . . .
4.2.2 Explanatory Variables . . . . . . . . . . . . . .
4.2.3 Application of the Expected Goals Model . . . .
4.3 Markov Game Model 1 . . . . . . . . . . . . . . . . . .
4.3.1 State Space: Context Variables and Field Zones .
4.3.2 State Transitions: Actions . . . . . . . . . . . .
4.3.3 Reward Function and Value Iteration Algorithm .
4.3.4 Valuing Individual Player Actions . . . . . . . .
4.4 Markov Game Model 2 . . . . . . . . . . . . . . . . . .
4.4.1 State Space: Context Variables and Field Zones .
4.4.2 State Space: Actions . . . . . . . . . . . . . . .
4.4.3 State Transitions . . . . . . . . . . . . . . . . .
4.4.4 Reward Functions and Value Iteration Algorithm
4.4.5 Valuing Individual Player Actions . . . . . . . .
4.5 Model Evaluation . . . . . . . . . . . . . . . . . . . . .
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19
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25
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27
29
31
32
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33
35
36
36
37
Evaluating Results
5.1 Expected Goals Model . . . . . .
5.1.1 Reliability . . . . . . . . .
5.1.2 Validity . . . . . . . . . .
5.1.3 Results . . . . . . . . . .
5.2 Markov Game Model 1 . . . . . .
5.2.1 Reliability . . . . . . . . .
5.2.2 Validity . . . . . . . . . .
5.2.3 Results . . . . . . . . . .
5.3 Markov Game Model 2 . . . . . .
5.3.1 Reliability . . . . . . . . .
5.3.2 Validity . . . . . . . . . .
5.3.3 Results . . . . . . . . . .
5.4 Case Studies . . . . . . . . . . . .
5.4.1 Mike Jensen . . . . . . .
5.4.2 Martin Ødegaard . . . . .
5.5 Reliability Out of Sample . . . . .
5.6 Evaluation of Research Questions
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39
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6
Conclusion
75
7
Recommendations for Further Research
77
Bibliography
79
Appendix
89
viii
A Tables from Tippeligaen
A.1 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
89
92
B Results from Expected Goals Model
B.1 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
96
98
C Results from Markov Game Model 1
101
C.1 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.2 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
D Results from Markov Game Model 2
109
D.1 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
D.2 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
E Markov Game Models: Validation from Team Evaluations
117
F Case Studies
119
F.1 Mike Jensen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
F.2 Martin Ødegaard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
G Results for 2016
G.1 Expected Goals Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G.2 Markov Game Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G.3 Markov Game Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
123
124
125
ix
x
List of Tables
2.1
EA Sports Player Performance Index EPL 2015/2016 . . . . . . . . . . . . . .
11
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Simple descriptive statistics for the dataset . . . . . . . . . . . . .
xG Model: Structure of shot data . . . . . . . . . . . . . . . . . .
xG Model: Variables considered in binary logistic regression . . .
Model 1: Context variables . . . . . . . . . . . . . . . . . . . . .
Model 1: A, action set for the team in possession of the ball . . .
Model 1: O, action set for the team not in possession of the ball .
Model 1: Structure of data . . . . . . . . . . . . . . . . . . . . .
Model 2: Context variables . . . . . . . . . . . . . . . . . . . . .
Model 2: Overview of the outcome parameter for the actions . . .
Model 2: Pass values for the context T P = 3, M S = 1, M D
1 and Z = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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=
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20
23
24
26
27
27
28
33
34
xG Model: Parameters for regular shots . . . . . . . . . . . . . . . . . . . . .
xG Model: Parameters for free kicks . . . . . . . . . . . . . . . . . . . . . . .
xG Model: Correlation coefficients of performance measures from 2014 to 2015
xG Model: All players scoring more than 8 goals in Tippeligaen 2015, ranked
by G/xG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model 1: Correlations to VG-børsen, Altomfotball and Transfermarkt in 2014 .
Model 1: Correlations to VG-børsen, Altomfotball and Transfermarkt in 2015 .
Model 1: Top 10 forwards, Tippeligaen 2015. Average total value = 0.1124,
Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . . . . . . . .
Model 1: Top 10 wingers, Tippeligaen 2015. Average total value = 0.1124,
Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . . . . . . . .
Model 1: Top 10 attacking midfielders, Tippeligaen 2015. Average total value
= 0.1124, Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . .
Model 1: Top 10 central midfielders, Tippeligaen 2015. Average total value =
0.1124, Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . . .
Model 1: Top 10 full backs, Tippeligaen 2015. Average total value = 0.1124,
Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . . . . . . . .
Model 1: Top 10 centre backs, Tippeligaen 2015. Average total value = 0.1124,
Minimum total value = -0.1248 . . . . . . . . . . . . . . . . . . . . . . . . . .
Model 2: Correlations between VG-børsen, Altomfotball, Transfermarkt and
Model 1 in 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
40
42
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
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= 0, H
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37
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57
xi
5.14 Model 2: Correlations between VG-børsen, Altomfotball, Transfermarkt and
Model 1 in 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 Model 2: Top 10 forwards, Tippeligaen 2015. Average total value = 0.4608,
Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Model 2: Top 10 wingers, Tippeligaen 2015. Average total value = 0.4608,
Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Model 2: Top 10 attacking midfielders, Tippeligaen 2015. Average total value
= 0.4608, Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . .
5.18 Model 2: Top 10 central midfielders, Tippeligaen 2015. Average total value =
0.4608, Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . . .
5.19 Model 2: Top 10 full backs, Tippeligaen 2015. Average total value = 0.4608,
Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 Model 2: Top 10 centre backs, Tippeligaen 2015. Average total value = 0.4608,
Minimum total value = 0.0059 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21 Model 1: Impact values for Mike Jensen 2015 . . . . . . . . . . . . . . . . . .
5.22 Model 2: Impact values for Mike Jensen 2015 . . . . . . . . . . . . . . . . . .
5.23 Model 1: Impact values for Martin Ødegaard 2014 . . . . . . . . . . . . . . .
5.24 Model 2: Impact values for Martin Ødegaard 2014 . . . . . . . . . . . . . . .
62
63
64
67
67
A.1
A.2
A.3
A.4
A.5
A.6
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89
90
91
92
93
94
B.1 xG Model: Correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 xG Model: All players scoring more than 8 goals in Tippeligaen 2014, ranked
by G/xG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 xG Model: Tippeligaen 2014 teams rated by xG difference . . . . . . . . . . .
B.4 xG Model: All players scoring more than 8 goals in Tippeligaen 2015, ranked
by G/xG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.5 xG Model: Tippeligaen 2015 teams rated by xG difference . . . . . . . . . . .
95
League Table Tippeligaen 2014 . . . . . .
Top 20 players with respect to goals 2014
Top 20 players with respect to assist 2014
League Table Tippeligaen 2015 . . . . . .
Top 20 players with respect to goals 2015
Top 20 players with respect to assist 2015
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C.1 Model 1: Top 10 players in offensive positions, Tippeligaen 2014. Average
value = 0.1118, minimum value = -0.2324 . . . . . . . . . . . . . . . . . . . .
C.2 Model 1: Top 10 players in defensive positions, Tippeligaen 2014. Average
value = 0.1118, minimum value = -0.2324 . . . . . . . . . . . . . . . . . . . .
C.3 Model 1: League table 2014 based on the model compared to the real table . .
C.4 Model 1: Top 10 players in offensive positions, Tippeligaen 2015. Average
value = 0.1124, minimum value = -0.1248 . . . . . . . . . . . . . . . . . . . .
C.5 Model 1: Top 10 players in defensive positions, Tippeligaen 2015. Average
value = 0.1124, minimum value = -0.1248 . . . . . . . . . . . . . . . . . . . .
C.6 Model 1: League table 2015 based on the model compared to the real table . .
58
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61
96
97
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102
103
104
105
106
107
D.1 Model 2: Top 10 players in offensive positions, Tippeligaen 2014. Average
value = 0.4668, minimum value = 0.0118 . . . . . . . . . . . . . . . . . . . . 110
xii
D.2 Model 2: Top 10 players in defensive positions, Tippeligaen 2014. Average
value = 0.4668, minimum value = 0.0118 . . . . . . . . . . . . . . . . . . . .
D.3 Model 2: League table 2014 based on the model compared to the real table . .
D.4 Model 2: Top 10 players in offensive positions, Tippeligaen 2015. Average
value = 0.4608, minimum value = 0.0059 . . . . . . . . . . . . . . . . . . . .
D.5 Model 2: Top 10 players in defensive positions, Tippeligaen 2015. Average
value = 0.4608, minimum value = 0.0059 . . . . . . . . . . . . . . . . . . . .
D.6 Model 2: League table 2015 based on the model compared to the real table . .
111
112
113
114
115
E.1 Model 1 and Model 2: p-values from the ADF test, Kendall’s tau and Spearman’s rho in 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
G.1 xG Model: Top 20 players 2016 with respect to G/xG . . . . . . . . .
G.2 Model 1: Top 20 players in all positions 2016. Average tota value =
minimum total value = -0.4143 . . . . . . . . . . . . . . . . . . . . .
G.3 Model 2: Top 20 players in all positions 2016. Average total value =
minimum total value = 0.0356 . . . . . . . . . . . . . . . . . . . . .
. . . . . 123
0.0752,
. . . . . 124
0.4541,
. . . . . 125
xiii
xiv
List of Figures
4.1
4.2
4.3
4.4
Data structure of F24 feed . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of passing data in terms of pass counts and passing accuracies . . .
Illustration of shot data in terms of conversion rates . . . . . . . . . . . . . . .
xG Model: Illustration of angle and length variables for a shot. Red lines show
the length and orange arrows show the angle. Angle measured in radians . . . .
4.5 Model 1: Field zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Model 1: Definition of a state . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Model 1: Illustration of states and transitions. Example from Tromsø - Sarpsborg 08 in 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Model 1: Zone values of state space for illustrative purposes. T P = 1, M D =
0 and GD = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Model 2: Field zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Model 2: Definition of a state . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.11 Model 2: Illustration of states and transitions. Example from Tromsø - Sarpsborg 08 in 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
xG Model: Scatter plot of the G/xG of all players with the correlation coefficient
between the 2014 and 2015 season . . . . . . . . . . . . . . . . . . . . . . . .
xG Model: ROC curve for the two shot models . . . . . . . . . . . . . . . . .
Model 1: Scatter plot and correlation between the two seasons . . . . . . . . .
Model 1: Scatter plots and correlation with Altomfotball . . . . . . . . . . . .
Model 1: Scatter plots and correlation with VG-børsen . . . . . . . . . . . . .
Model 1: Scatter plots and correlation with Transfermarkt . . . . . . . . . . . .
Model 2: Scatter plot and correlation between the two seasons . . . . . . . . .
Model 2: Scatter plots and correlation with Altomfotball . . . . . . . . . . . .
Model 2: Scatter plots and correlation with VG-børsen . . . . . . . . . . . . .
Model 2: Scatter plots and correlation with Transfermarkt . . . . . . . . . . . .
Model 1: Box plot for Mike Jensen in 2015 excluding shots . . . . . . . . . . .
Model 2: Box plot for Mike Jensen in 2015 . . . . . . . . . . . . . . . . . . .
Model 2: Comparison of Mike Jensen vs league for 2015 . . . . . . . . . . . .
Model 1: Box plot for Martin Ødegaard in 2014 excluding shots . . . . . . . .
Model 2: Box plot for Martin Ødegaard 2014 . . . . . . . . . . . . . . . . . .
Model 2: Comparison of Martin Ødegaard vs league for 2014 . . . . . . . . . .
xG Model: Scatter plot and correlation of G/xG between 2015 and 2016 . . . .
20
21
22
23
26
27
28
31
33
34
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42
43
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63
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xv
5.18 Model 1: Scatter plot and correlation between 2015 and 2016 . . . . . . . . . .
5.19 Model 2: Scatter plot and correlation between 2015 and 2016 . . . . . . . . . .
70
70
E.1 Model 1 and Model 2: Real table position compared to estimated each matchday 117
F.1
F.2
F.3
F.4
F.5
xvi
Model 1:
Model 1:
Model 2:
Model 1:
Model 1:
Box plot for Mike Jensen 2014 including shots . .
Box plot for Mike Jensen 2014 excluding shots . .
Box plot for Mike Jensen 2014 . . . . . . . . . . .
Box plot for Mike Jensen in 2015 including shots .
Box plot for Martin Ødegaard 2014 including shots
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119
120
120
121
121
Chapter
1
Introduction
Analysing the competition has for a long time been an important task in the world of association
football (throughout this thesis referred to as football). Typical applications of such analyses
are to get information and insights on opponents before, during and after matches, to keep
track of performance of own squad and to scout for talent. Analyses have previously been
qualitative in nature. However, quantitative data analysis has gained momentum in recent years,
and several companies, like Opta Sports and Prozone, have made a business out of collecting
fine grained data during football matches. This data is sold to customers such as football clubs,
betting companies, TV broadcasters and newspapers. In combination with the steady increase
in computational power, these data sets create new possibilities for statistical analysis in sports,
commonly referred to as sports analytics. Football clubs are attempting to lever the potential
for valuable information that data and statistical analysis can provide, and several of the biggest
clubs in the world try to exploit this potential to create and sustain a competitive edge. The
onset of sports analytics is also apparent amongst smaller clubs who are not able to compete
financially with the larger ones. They are trying to close the gap in resources through innovation
within analysis and player logistics. Good examples of such clubs are Brentford FC in the
English Championship and FC Midtjylland in the Danish Superliga. Brentford FC has moved
from English League Two (fourth tier in England) in 2009, to almost being promoted to the
English Premier League in the 2014/2015 season (losing to Midlesborough in the playoffs). In
the 2014/2015 season FC Midtjylland won the Danish Superliga for the first time in club history.
Both these clubs claim that sports analytics has been an important part of their culture, and an
important factor in their recent success.
Several areas of application are possible for statistical analysis on data from association
football. The most popular ones in academic research include identifying key performance
indicators (KPIs), analysing the home field advantage, finding the effect of situational variables
and investigating set-pieces, crosses, yellow and red cards and referee bias. Such analyses have
been performed for several years. More recent analyses focus on fine grained tracking data and
evaluation of individual players and their involvements on the field.
Evaluation of player actions on the field has for many years been quite unsophisticated. It
has mostly been based on simple parameters, like goals scored, the number of assists and passing accuracy for attacking players and goals conceded, duels won and number of interceptions
for defending players. This approach has several drawbacks. First of all, the context of where
the players perform actions on the field is not incorporated, and players are evaluated based on
absolute measures rather than compared to expectations. It is a better accomplishment to score
1
Chapter 1. Introduction
on a well placed free kick from 25 meters than to convert a penalty kick. Similarly, for defensive
players, an interception should be valued higher if it prevents a good scoring opportunity.
In The Numbers Game (Anderson and Sally (2013)), the authors emphasise two other factors that the traditional valuation does not take into account when valuing football players. The
first is the importance of the goals. That is, how many points that are ensured by the goals a
player scored. A match winning goal is more valuable then scoring goal number five late in
the game in a 5-0 victory. Secondly, defensive players seems to be undervalued relative to offensive players. Of the 20 most expensive transfers in football history, only one is a defender
(David Luiz, 50 million GBP, from Chelsea to Paris Saint-Germain in 2014). The rest are either
offensive midfielders or attackers. Data analysis shows that this is an irrationality and a market
inefficiency, as it is stated in Anderson and Sally (2013):
“Through the ten Premier League seasons from 2001/2002 to 2010/2011 it appears
that not conceding any goals during a match ensured almost 2.5 points per match.
Compared with scoring one time, which gives approximately one point per match,
it is twice as valuable to prevent the opponent scoring a goal. In fact, conceding
only one goal during a match gives the team 1.5 points on average, approximately
30 % more than the value of scoring only one goal.”
As mentioned, quantitative analyses has gained momentum in recent years. The increasing
body of research has to a certain extent rectified several biases and challenged many irrationalities in the game of football. One popular concept that has been useful in this effort is expected
goals (xG). This is a concept that incorporates the context of the player by including several
variables affecting a shot. It assigns a likelihood for converting a shot into a goal. In this way,
the model can provide a more objective foundation for evaluation of players by looking at how
they performed compared to expectations. However, xG models only look at shots, and their
best application is therefore to evaluate the efficiency of primarily attacking players. xG models
in their traditional form are not able to assign values to actions that leads to a shot. For example, the assist to a goal could be a brilliant penetrating pass, but the passing player will not
get a share of the xG value in such models. In recent years, efforts have been made in order to
evaluate all player involvements, not only shots.
1.1
Motivation
Valuing player involvements in association football can be a cumbersome and difficult task.
Besides favouring players in offensive positions, player evaluations are very often influenced
by human bias. The main objective in this thesis is to extend the existing body of research on
football with models that can objectively evaluate individual player involvements. The obtained
values can form a basis for player ratings free from human bias, which could serve several
purposes for clubs. A substantial amount of resources is committed in scouting for talent, and
the objective player ratings can be used to make short lists of players that are worth a closer
examination, both domestically and internationally. This could make the process of scouting
more accurate and less time consuming, which can have a big monetary impact in the long run.
They can also be used to scout own squads in order to monitor the performance of the players
and see which players are performing on the desired level.
2
1.2 Research Questions
Another motivation is to take on challenges regarding modelling of contextual parameters in
football and the rating of defensive players. This is, to the authors knowledge, only addressed
to a limited extent, and little work has previously been done on Norwegian football.
An objective and data driven player evaluation can also create possibilities for a financial
analysis, useful for coaches, directors, owners and other stakeholders in football clubs. They can
utilise such models either for evaluating contracts or values of players on the transfer market.
They operate in an environment where competition for the best players and the best talent is
increasingly fierce. Crucial decisions regarding player logistics and transfers are being made
every day. Getting it right more often than competitors can create competitive advantages, and
may be an important step to ensure success in the long run.
1.2
Research Questions
The purpose of this thesis can be synthesised as incorporating contextual parameters and situational variables when assessing player performance, developing a player efficiency rating, and
rate outfield players based on all involvements. For clarity and guidance, a set of formal research questions (RQs) are developed. They are used to guide the work done in this thesis, and
are addressed in the discussion of results and conclusion regarding how successful the works in
this thesis have been. The three formal research questions are formulated as:
RQ 1 Is it possible to create a statistically significant xG model that assesses the quality of all
shots in Tippeligaen in order to evaluate the efficiency of primarily offensive players?
RQ 2 Is it possible to create a Markov game model for football that is able to evaluate all player
involvements in a match and rate players over the course of a season?
RQ 3 Is it possible to reveal undiscovered talent or identify under- and overvalued players
based on the evaluation of individual player involvements?
1.3
Limitations
To conduct a relevant and applicable analysis of individual player performance is far from a
trivial task. Abilities of clubs to gather the right kind of players with the correct set of skills and
complementary characteristics is crucial in order to succeed. In football, as a collective sport,
the total ability of a team is greater than the sum of the individual parts. These complementary
effects are not accounted for in the models developed in this thesis. Neither are effects that
come from team configurations and stability of playing style and strategy. Performing on a high
level is possibly easier when the coach is the same across many seasons.
Another limitation in this thesis is that all players are evaluated on the same basis. Obviously, different skills or characteristics are important for a defender and a forward. The nature
of the data set used in this thesis is such that it makes it harder to appreciate defensive attributes
than offensive. This makes the models biased towards attackers and offensive involvements.
Only data from two seasons of football is used. Although the two seasons include a high
number of events, it would have been preferable to have more data in order to evaluate the
validity and consistency of the models developed. In addition, some events have been removed
by the authors, which has resulted in a further reduction of the data set. These events are mainly
events which does not involve the ball.
3
Chapter 1. Introduction
Because of the novelty of modelling football as a Markov game, the validity of the models
in this thesis is hard to assess. The only appropriate comparable sources found, are subjective
player ratings and estimated market values. The authors admits that using only correlation
coefficients between these benchmarks and the models as a basis for validation, is questionable.
1.4
Report Structure
Chapter 1 serves as an introduction and presents the motivation, three research questions and
the limitations considered in this thesis.
Chapter 2 gives an overview of the existing literature on sports analytics, where the main focus
is on relevant works regarding xG models, Markovian approaches and player evaluation.
Chapter 3 describes the theoretical background for the models developed in this thesis.
Chapter 4 introduces the data set, and explains the procedures for building the models in this
thesis.
Chapter 5 presents the reliability, validity and results from the different models, which are
further discussed with regard to the research questions.
Chapter 6 highlights the major findings, and conclusions are drawn regarding the merits of
this research.
Chapter 7 suggests extensions of the models, and addresses the potential for further research.
4
Chapter
2
Literature Review
The focus on sports analytics related to decision making has for many years experienced a significant increase, also in the world of football. Important stakeholders like club owners, managers and analysts have realised the importance of using data driven arguments when analysing
one of the most complex and dynamic sports in the world. This is also reflected in the growth of
the body of research on football in recent years. Due to the prematurity of this research field, the
first part of this chapter gives an overview of important research. The second part investigates
the work performed on the concept of expected goals (xG), and the third part looks at Markov
game models used within sports analytics. The fourth part focuses on research performed to
evaluate individual player involvements and player valuations. Analyses on valuing football
players financially is the focus of the fifth and last part of this chapter.
2.1
Sports Analytics in Football
It is hard to define exactly when the field of sports analytics was conceived, but Charles Reep
is by many viewed as a pioneer within football analytics. In Reep and Benjamin (1968) and
Reep et al. (1971) the passing patterns of English Premier League clubs are analysed, and the
research concludes that “It seems, however, that chance does dominate the game and probably
most similar ball games”. It is now widely recognised that luck is a considerable element during
a football match, but also that regularities can be revealed and influenced. From the beginning
of sports analytics, a wide array of approaches on research topics and methods has been used in
the quest for revealing these regularities, and by examining 140 journals in operations research,
statistics, applied mathematics and applied economics, Coleman (2012) defines the field of
sports analytics to be “sizable and growing”.
One distinctive research topic that is pronounced in the investigated literature is that of performance analysis and finding key performance indicators (KPIs). The main focus of such analyses is team performance, and KPIs like passes, ball possession and ball recoveries are popular
subjects. Passing is a cardinal event in a football match, and it is interesting to see that different
analyses give different results. While Reep and Benjamin (1968) and Wright et al. (2011) find
that short passing sequences are favourable, Hughes and Franks (2005) and McHale and Scarf
(2007) find that longer passing sequences are preferable. These findings further emphasise the
notion that one should take care when interpreting such analyses. Another KPI that has gotten
a lot of attention is ball possession. There are studies performed that indicate ball possession
as a statistical significant parameter that affects the outcome of matches (Lago-Ballesteros and
5
Chapter 2. Literature Review
Lago-Peñas (2010), Castellano et al. (2012), Collet (2013), Bradley et al. (2013), Bjertnes et al.
(2015)). However, ball possession is a complex parameter, which is affected by many other
parameters. Lago and Martı́n (2007) finds that 66 % of the variation in ball possession in the
2003/2004 La Liga is determined by the situational variables team strength, match location
and match status. Two other KPIs that are given some attention are ball recoveries (Barreira
et al. (2014)) and penalty area entries (Ruiz-Ruiz et al. (2013)), which both reached statistical
significance.
Situational variables has gotten a lot of attention by the sports analytics community in recent
years. Lago-Peñas (2012) sums up the concept of situational variables concisely:
“Given that soccer is dominated by strategic factors, it is reasonable to suggest that
situational variables of match status (i.e. whether the team is winning, losing or
drawing), strength of opposition (strong or weak), and match location (i.e. playing
at home or away) may somehow influence the teams’ and players’ activities. These
situational variables need to be analysed in depth to understand their influence in
team sports.”
It is widely recognised in both the football and the sports analytics community that these
variables influence other in-match events. Several articles support this, and amongst the parameters that seem to be affected are technical behaviour (Taylor et al. (2008)), distance covered by
single players (Lago et al. (2010)), ball possession (Lago-Peñas and Dellal (2010)), attacking
patterns (Machado et al. (2014)) and ball recovery dynamics (Almeida et al. (2014)). Thus,
there is substantial evidence that situational variables should be accounted for when analysing
other parameters. One situational variable that has gotten especially much attention is the home
field advantage. Within this narrow scope, most analyses support the notion of home field advantage and its affect on other parameters (Clarke and Norman (1995), Brown Jr et al. (2002),
Bray et al. (2003), Carmichael and Thomas (2005)). Such parameters include team strategy,
shot statistics and disciplinary behaviour (Seçkin (2006), Seçkin and Pollard (2008)). Goumas
(2014) also shows that travelling time and length negatively affect the away team, which enhances the home field advantage. By using six seasons from 72 countries, Pollard et al. (2008)
summarises the widespread the home field advantage neatly:
“In Europe, home advantage in the Balkan countries, especially Bosnia and Albania, is much higher than average. It is generally lower than average in northern Europe, from the Baltic republics, through Scandinavia to the British Isles. In South
America, home advantage is high in the Andean countries and lower elsewhere,
especially in Uruguay.”
The displacement patterns and tactical patterns of a football team are also distinct features
which have been analysed by several researchers. Displacement patterns refer to how a team
is geometrically organised on the field, while tactical patterns can refer to how a team passes,
how they put pressure on the opponent and which playing style they prefer. Much of the analyses on displacement patterns are quite new due to the requirement of advanced tracking data
on the players in order to obtain sound analyses. Several analyses find that the organisational
shape and structure of the team are significantly different dependent on the type of play (Moura
et al. (2012), Duarte et al. (2012), Duarte et al. (2013), Barreira et al. (2013)). Bialkowski et al.
(2014a) uses spatiotemporal data (containing details on both space and time) to perform automatic formation analysis to investigate the difference in team behaviour when playing home
6
2.1 Sports Analytics in Football
and away. This is further used in Bialkowski et al. (2014b) to identify the uniqueness in team
passing, movement and interaction. Tactical patterns have also been investigated, and the analyses find that attacks differs dependent on whether the team is winning or losing (Jankovic
et al. (2011), Niu et al. (2012)). In recent years, some pioneering work within analysing passing
strategies have been done. Gyarmati et al. (2014) uses automatic extraction of passing strategies
to search for a unique passing style in football, and reveals for instance that Barcelona’s tikitaka “does not consist of uncountable random passes but rather has a precise, finely constructed
structure”. Also in analyses of tactics, spatiotemporal data is used. Niu et al. (2012) applies
trajectory extraction to analyse football tactics, and finds six distinct attack patterns. Bojinov
and Bornn (2016) studies team tactics in the English Premier League. This analysis investigates
the ability of each team to control the ball in offensive zones, and their ability to disrupt the
opponents flow of play, which induce possibilities for tactical considerations by managers.
Surprisingly, little research performed on free kicks and corner kicks was found (Bray and
Kerwin (2003), Taylor et al. (2005), Alcock (2010), De Baranda and Lopez-Riquelme (2012)).
Penalties on the other hand, being a monumental occurrence in a football match, are a more
popular subject for analysis. Lopes et al. (2008) and Jordet and Hartman (2008) investigate
the psychological aspect of a penalty and find a link between player behaviour and outcomes
of penalties. Other analyses are more quantitative in nature. Bakkerode and Koning (2014)
reveals that the number of penalty kicks and the outcome are positively affected by the home
field advantage, Misirlisoy and Haggard (2014) finds that the diving direction of the goalkeeper
is affected by the preceding penalty and Noël et al. (2015) uncovers three variables predicting
the penalty kick strategy: attention to goalkeeper, the run-up and the kicking technique. Other
important in-match events that have been analysed for their impact on football match results are
crosses (Kumar et al. (2013), Orth et al. (2014), Yamada and Hayashi (2015), Vecer (2014)),
yellow cards (Unkelbach et al. (2008), red cards (Ridder et al. (1994), Vecer et al. (2009),
Mechtel et al. (2011)) and referee bias (Buraimo et al. (2010), Buraimo et al. (2012), Reilly and
Witt (2013), Constantinou et al. (2014), Goumas (2014)).
This thesis utilises data from Tippeligaen, and it is therefore interesting to investigate what
research related to Norwegian football that exists. Brillinger (2007) is a piece that looks at
the 2003 season of Tippeligaen, and based on the number of goals estimates the likelihood for
winning, drawing or loosing. Tenga et al. (2010a) applies logistic regression on team possession
from Tippeligaen 2004. The purpose of this study is to examine what effects match location
has on playing tactics. The data set used in this article is further exploited in Tenga et al.
(2010b), Tenga et al. (2010c) and Tenga et al. (2010d). These three analyses also apply logistic
regression. The first one investigates the effect playing tactics has on entries into the a defined
zone near the opponents goal, while the second one looks at the effect playing tactics has on
goal scoring. The third one assesses the effectiveness of teams by studying the relationship
between broader measures (scoring opportunities and offensive zones possessions) and goals
scored. Halvorsen et al. (2013) presents a case study including a video sensor system installed
at Alfheim stadion. However, this system seems to be more appropriate for qualitative analysis
than as a basis for quantitative modelling. Sæbø (2015) looks at several leagues in his analysis,
including fixtures from six seasons (2009 - 2014) of Tippeligaen. This analysis uses the financial
standings of the clubs as the point of origin, and valuates players based on how they affect the
revenue of the club. This work is synthesised in Sæbø and Hvattum (2015).
To the authors knowledge, the specialisation project Bjertnes et al. (2015) is the first analysis
that utilises data consisting of descriptive match statistics in order to measure team performance
on a quantitative basis. This work applies linear and logistic regression in order to determine
7
Chapter 2. Literature Review
which match statistics that were prominent in Tippeligaen 2015, and how they affected the
likelihood for winning, drawing or losing a match. Although having a focus on Norwegian
football, none of the above analyses examine individual player performance based on in-match
performance statistics. Thus, this thesis offers new insights on Norwegian football by focusing
on the performance of individual players.
2.2
Expected Goals (xG)
In recent years, the concept of xG has gotten a lot of attention in sports analytics communities.
The largest community is blogs and fan pages online, where a variety of models are presented.
However, also in academic circles, xG has existed for some years within several sports. In
both NBA (National Basketball Association) and NHL (National Hockey League) this concept
has been embraced as a useful metric. xG is a metric that incorporates contextual parameters
when evaluating a shot. Commonly used parameters are distance to goal, angle on goal, which
body part that was used when shooting, if the assist was air borne or not and the number and
proximity of defenders. Hence, in contrast to more common metrics like the number of shots
or shooting accuracy, the xG metric also evaluates the quality of the shots.
xG has been the topic of several research papers. One that uses the concept explicitly is
Macdonald (2012). This article analyses NHL teams and players over three seasons, and develops an xG model using ridge regression with goals, shots, hits, hits against and faceoffs as
explanatory variables. When compared to simpler models, this model shows a higher correlation between actual and predicted goals. It also has the lowest mean squared error, and is
therefore considered more appropriate for evaluating individual players.
The research paper discussed above uses discrete in-match events, but another possibility is
to use fine grained tracking data. The provision of such data has increased in recent years and
works have been performed to evaluate the feasibility of utilising such data for football analysis (Kim et al. (2011), Bialkowski et al. (2014c), Gudmundsson and Wolle (2014)), assessing
football tactics (Wei et al. (2014), Fernando et al. (2015)) and identifying regularities within
football matches (Lucey et al. (2013a), Bialkowski et al. (2014b)). Furthermore, tracking data
has been used to generate xG models in other sports than football. Chang et al. (2014) exploits
the opportunity of using a tracking data system in the NBA to analyse and quantify shooting.
The most popular metric used in NBA for evaluating the shooting ability of players and teams
is Effective Field Goal Percentage (eFG%), which is a ratio of the number of field goals to the
number of field goal attempts for a player. However, this paper recognise the need for incorporating the quality of a shot into a metric: “If the player is standing still with no one within
10 feet, that is an entirely different shot than if the player is fading off the dribble with two
defenders in his face.” Variables like shot distance, shot angle, defender distance, defender angle, player speed and player velocity angle are used as input parameters in the model. Machine
learning methods are then utilised on the data set, and the author establishes two new metrics
for basketball: Effective Shot Quality (ESQ), which capture the quality of the shot and eFG%+,
which is ESQ subtracted from eFG%, and shows how much better than expectation a player or
a team shoots.
In the world of football, Lucey et al. (2014) uses a season worth of tracking data with 10,000
shots from a professional league provided by Prozone to quantify the likelihood of scoring
by specifying several input parameters. The analysis is complex, dividing the shots into six
different match contexts: open play, counter attacks, corners, penalties, free kicks and set pieces.
8
2.3 Markov Models in Sports Analytics
The shots are analysed in the context of strategic features like defender proximity, interaction
of surrounding players and speed of play. Then the authors use logistic regression to estimate
the xG for each shot in order to rate teams based on their efficiency.
As can be seen from the works above, several academic pieces apply the concept of xG.
It is recognised that xG is a useful measure for evaluating the quality of the shots to better
examine if the team or player performed well. No academic works were found that present xG
models based on data from Tippeligaen. This thesis presents an xG model, which is developed
by applying logistic regression. However, tracking data from Tippeligaen is not available, thus
event-based data is utilised. Nonetheless, to the authors knowledge, the xG Model presented
in this thesis is the first of its kind for Tippeligaen, and an important step towards developing
quantitative models for player performance evaluation in Norway.
2.3
Markov Models in Sports Analytics
As can be seen in the book by Wright (2015), on applications of operational research methods
in sports, it is not uncommon to apply Markov processes when analysing sports phenomena.
Hirotsu and Wright (2002) presents a continuous Markov process with four states. These states
are when team A scores a goal, team A is in possession of the ball, team B is in possession of the
ball and when team B scores a goal. As stated in the paper, defining these states and calculating
the transition probabilities “makes it possible to estimate the probability distributions of goals
scored and the expected number of league points gained, from any position in a match, for
any given set of transition probabilities and hence in principle for any match”. By knowing
these time dependent parameters, this paper estimates the optimal time for tactical changes
during a match, like player substitutions. In Hirotsu and Wright (2003a), a similar model is
used to evaluate the offensive and defensive strengths of a football team. These results are then
utilised to form a substitution strategy for the team, showing how many of each type of players
should start and be substituted during a match. The same Markov model is applied again in
Hirotsu and Wright (2003b). This analysis is similar to the former, but focuses on visualising
team characteristics like offensive and defensive strength, the home field advantage and relative
success against particular teams.
Markov models have also been developed in ice hockey. Thomas (2006) defines 19 states
specified by puck possession, takeaways, faceoffs and goals. Using this state space he models
ice hockey as a semi-Markov process, which is used to determine the average number of goals
scored by a team as a function of the starting state. These values are then used to look at which
tactic that should be used in a certain situation. An interesting finding is that giving up puck
control in order to get territorial advantage is beneficial both defensively and offensively and,
as he states “may explain the prominence of location-based defence throughout professional
hockey”.
It can be seen that none of the analyses above are related to individual player performance
in order to rate players, but looks to aid teams in tactical decisions. Thomas et al. (2013) marks
a start of using Markov process methodology to evaluate individual players. In this analysis,
scoring rates for each NHL team are modelled by a semi-Markov model. However, every player
will in some way affect this scoring rate, and hazard function models are used for quantifying
this effect, enabling player rating within different player positions.
Also in basketball the Markovian approach has been used. Cervone et al. (2014) present an
analysis that uses tracking data to value player decisions in real time. The Markovian element in
9
Chapter 2. Literature Review
this analysis is the assumption that the decision of the ball handler depends only on the current
spatial configuration of the team, disregarding the play history. Statistical methods are used
for computing a new metric, expected possession value (EPV). EPV is a quantification of the
decision a player makes, and if the decision leads to a higher or lower probability of scoring.
This measure is further used to create expected possession value added (EPVA), that is, how
much better decision a player takes compared to league average. This is a cutting edge analysis,
which utilises complex statistical algorithms on fine grained tracking data.
A thesis highly relevant to this thesis is Routley (2015), which has been a source of inspiration. He looks at multiple seasons in NHL, including 600,000 play sequences and 2.8 million
events to model ice hockey dynamics as a Markov game between two teams. A large state space
is created, where a state is defined by three context variables (time period, goal differential and
manpower differential) and an action history in the form of a play sequence. The sequences are
made by defining start and end events, which suits ice hockey dynamics. Every time an action
is taken by a player, a state transition is initiated. Multiple reward functions are used to obtain
the impact values for goals, wins and penalties. The value iteration algorithm is run over the
Markov game state domain to assign a value to each state depending on what is rewarded in the
state space. A valuing of individual player involvements is done by looking at the aggregated
impact by each player for each season.
In the literature review above no analyses were found on Norwegian football, nor on individual football player performance, that utilises Markovian techniques. Thus, to best of the authors
knowledge, the Markov models presented in this thesis are the first in Norway, and probably the
first of their kind published in the football analytics community.
2.4
Evaluation of Individual Player Performance
In recent years, focus has shifted towards evaluation of individual players, their involvements
and their contribution to the team. Due to the use of simple statistics, several in-match phenomena are not captured, like an inactive defence on a superior team, full backs that need to adjust
their roles depending on in match situational variables and the context a player was in when
making a choice and performing an action. However, new methods for measuring performance
and fine grained data sets now make it possible to incorporate much more information. This
enable evaluation of players that perform well, but ”disappear” in such simple statistics.
The EA Sports Player Performance Index is an index that rates players in the two top tiers
of football in England - the Premier League and the Championship. This index was developed
due to the desire from the football leagues and a news media company to have a rating system
that was built on a statistical basis, which enables comparison of players in different positions.
In McHale et al. (2012) this index is described, and it consists of six subindices from which
individual players get scores: match contributions, winning performance, match appearances,
goals scored, assists and clean sheets. Based on these parameters, the performance index is
created, and the top 10 players of the English Premier League in the season of 2015/2016 can
be seen in Table 2.1. (Premier League (2016))
Although being based on statistical modelling, this index only accounts for simple statistics
that does not capture important dynamics of player involvements. Also, the situational context
for where the action was carried out is not incorporated.
Another analysis that evaluate players based on simple statistics is that of Santı́n (2014).
He measures ”technical efficiency” of Real Madrid legends by using four variables collected
10
2.4 Evaluation of Individual Player Performance
Table 2.1: EA Sports Player Performance Index EPL 2015/2016
Rank
1
2
3
4
5
6
7
8
9
10
Player
Club
Harry Kane
Tottenham Hotspur
Riyad Mahrez
Leicster City
Jamie Vardy
Leicster City
Sergio Aguero
Manchester City
Mesut Özil
Arsenal
Christian Eriksen Tottenham Hotspur
Romelu Lukaku
Everton
Odion Ighalo
Watford
Dimitri Payet
West Ham United
Alexis Sánchez
Arsenal
Position
Forward
Midfielder
Forward
Forward
Midfielder
Midfielder
Forward
Forward
Midfielder
Forward
Index Score
1026
957
938
770
769
732
711
700
691
674
from the official Real Madrid website: number of official games played, the number of titles
won with their national team, the number of international titles won with Real Madrid and the
number of goals scored. The method of Data envelopment analysis (DEA) is applied to find
the most efficient Real Madrid legend since 1946 through 2011. This model succeeds in rating
players based on their historical performance, but does not provide insights on how to measure
and rate players based on in-match choices, actions and events.
In the period after 2010, it seems that large efforts have been made using in-match events
to generate statistical models that capture important dynamics. Macdonald (2010) develops a
regression based model to evaluate individual ice hockey players. A similar analysis is also
performed in the NBA, by Fearnhead and Taylor (2010). These models are called plus-minus
models, which enable a player rating based on how the teams perform when the player is on
the ice/field as opposed to when he is not. The model from NHL is improved and adjusted in
Macdonald (2011a) and Macdonald (2011b), and so far this particular research has ended in
an xG model which is described in Macdonald (2012) (see Section 2.2). The output of this
model is a list of players and their performance measured in xG per 60 minutes. This analysis is
refined in Macdonald et al. (2012), where both goalies and skaters are evaluated, in addition to
NHL teams. Another analysis with data set from NHL is Macdonald et al. (2013). In this piece,
a playmaker metric is created, which enables the identification and evaluation of playmakers,
and their contribution to the team. This is an important and interesting analysis, because earlier
works focused mainly on the actual scoring attempt and the quantification of a chance, and
the probability of converting it into a goal. This analysis looks at the events that lead up to a
goal, that is, quantifies the steps preceding the scoring attempt. This gives valuable insights
regarding player contribution besides scoring goals and providing assists. Schuckers and Curro
(2013) builds on previous work and creates a model that rates both forwards and defenders. The
model is built with on ice events from several NHL seasons, and the method of ridge regression
is applied. This methodology ensures a probability assigned to each action event. The per event
value is converted into a per season estimate of the number of wins created per player, which is
called Total Hockey Rating (THoR).
The models described above are important tools when trying to understand and recreate the
complex dynamics of sports. However, there are some limitations that are not accounted for.
One important obstacle is to incorporate match status. Pettigrew (2015) bypasses this challenge
by creating a model that generates in-game winning probabilities. This is a time line where the
probability for one of the teams winning fluctuates dependent on events during the game. As he
11
Chapter 2. Literature Review
describes himself:
“...imagine that one of those players scored all of his goals in overtime, while the
other player scored all of his when his team already had a large lead. It is clear that
the first player is a more valuable asset. But what about in situations that are not
nearly as clear-cut? By using the change in win probability when a goal is scored,
we can evaluate how much a player’s goal contributions impact their team’s chance
of winning the game.”
Thus, this analysis assesses the offensive efficiency of NHL players, which can be valuable
when investigating players with identical goal scoring rates.
Other sports have also been analysed with regards to individual player evaluation. Singh
and Ahmad (2015) looks at land hockey, and applies machine learning techniques to predict
the better performing player. A recent article that looks at movement patterns in football is
Gyarmati and Hefeeda (2016). Here, an event-based data set from Spanish La Liga 2012/2013
is used to extract player movements throughout the season. Vectors are constructed and contain
information on start and end of a movement including time stamps, the speed of the movement
and whether the player was in possession of the ball or not. These vectors are used for extracting
movement characteristics for each player in the league. The uniqueness and consistency of
player movements are quantified, which enables a comparison of players.
The above descriptions show that the focus on individual player performance in recent years
has increased. The quantitative methods applied on sports leagues like NHL and NBA are increasing in complexity. Furthermore, it seems like the football community has understood the
importance of a numerical foundation for aiding decision making regarding individual players. However, in Norwegian football, no analyses were found on quantifiying individual player
performance, which is the goal of this thesis.
2.5
Financial Valuation of Football Players
Surprisingly, not much academic work was found on the task of valuing football players financially. Especially notable is the missing research on the link between a players performance on
the team and his market value. With the amount of money circulating in the world of football,
it is reasonable to believe that there are several stakeholders looking for inefficiencies in the
pricing of football players. Why this is the case is hard to determine, but one reason can be the
difficulty of pricing intangible assets like human beings. Another possible explanation is that
clubs are not disclosing their models to the public.
However, some studies exist. An early work by Amir and Livne (2005) looks at player contracts, and finds no evidence between the investment in player contracts and future benefits for
their clubs. In Tunaru et al. (2005) this link is better established by developing a framework for
pricing football players based on performance data provided by Opta. Real options theory is
used to model the uncertainty surrounding the performance of football players. The framework
developed in this paper is also used in Tunaru and Viney (2010), giving a more thorough presentation of the application of the real option pricing formulas.
It can be seen from the above literature review that sports statistical analysis has been through
an evolution. The different methods have existed for some time, but the increasing quality of
the data sets opens up to novel approaches. Now, advanced methods and models are applied on
12
2.5 Financial Valuation of Football Players
both tracking data and large event based data sets. These give valuable insight into sports with
complex dynamics. This thesis seeks to further develop the field of sports analytics, and inspire
to further innovations to improve decision making in the game of football.
13
Chapter 2. Literature Review
14
Chapter
3
Basic Theory
This chapter presents the basic theory behind the methods applied in this thesis. First, the theory
behind binary logistic regression is described, then the definition, notation and dynamics of a
two-agent Markov game are introduced. Application of the theory is described in Chapter 4.
3.1
Binary Logistic Regression
Logistic regression is a statistical technique useful when the dependent variable is discrete instead of continuous. The goal is to find the best fitting model to describe the relationship between a dependent (outcome) variable and a set of explanatory (independent) variables. Logistic
regression makes it possible to associate likelihoods to outcomes of discrete random variables.
The explanatory variables can be dichotomous (taking only two values), polychotomous (taking
k > 2 values) or continuous. When the dependent variable only has the two outcomes 0 and 1
it is called binary logistic regression. This is appropriate when modelling football phenomena
because of dependent variables like match outcome,
⇢
1, if a team is winning match i
yi =
(3.1)
0, otherwise
or the outcome of a shot,
yi =
⇢
1,
0,
if shot i resulted in a goal
otherwise
(3.2)
and to assess what explanatory variables that significantly influence these outcomes.
Following Hosmer and Lemeshow (2000) and Wikipedia (2016), assume a series of N observed data points, where each data point i includes a set of m explanatory variables and an
associated binary outcome variable Yi . Also assume that Yi is Bernoulli distributed with an
unobserved probability ⇡i of outcome 1. This is specific for the outcome of observation i, but
assumed to somehow relate to the explanatory variables. Hence, the expected value of Yi given
the explanatory variables is equal to ⇡i , or E{Yi |x1,i , . . . , xm,i } = ⇡i .
The basic idea of logistic regression is the same as for linear regression, which is to model
a dependent variable (in this case the probability ⇡i ) using a linear combination of explanatory
variables by estimating a set of regression coefficients. The binary logistic regression is run on
the logit of the unobserved probabilities, which are defined as
15
Chapter 3. Basic Theory
logit(⇡i ) = ln
✓
⇡i
1
⇡i
◆
=
0
+
1 x1,i
+ ... +
m xm,i ,
(3.3)
and the estimated probabilities are calculated by solving Equation (3.3) for ⇡i , which yields an
expression for the inverse of the logit as
E{Yi |x1,i , . . . , xm,i } = logit 1 ( ˆ0 + ˆ1 x1,i + . . . + ˆm xm,i )
=
1
1+e
(3.4)
( ˆ0 + ˆ1 x1,i +...+ ˆm xm,i )
where ˆ0 , ˆ1 , . . . , ˆm are the regression coefficients obtained by running the regression.
In order to validate a binary logistic regression, it is possible to look at a classification
table. A cutoff or threshold must be specified to determine if an outcome of the dependent
variable is classified as 1 or 0, and the hit rate can be calculated by looking at how many
observation the model got correct. Following Hosmer and Lemeshow (2000), the sensitivity
of the model is defined as the probability of predicting an outcome as positive when Y = 1,
while the specificity is defined as the probability of predicting negative when Y = 0. Only
looking at those measures gives limited insight because a threshold has to be specified. A
more comprehensive description of classification accuracy is given by the area under the ROC
(Receiver Operating Characteristic) curve, which assesses the discrimination abilities of the
model. It plots the probability of detecting true observation (sensitivity) and false observation
(1-specificity) for an entire range of possible cutoff points.
Hosmer and Lemeshow (2000) also provide a general rule for values of the area under the
ROC curve, which is between 0 and 1, as
ROC = 0.5 : this suggests no discrimination (that is, one might as well flip a coin)
0.7  ROC < 0.8 : this is considered acceptable discrimination
0.8  ROC < 0.9 : this is considered excellent discrimination
ROC 0.9 : this is considered outstanding discrimination.
3.2
Markov Games
The idea behind a Markov game, also called a stochastic game, is to model two or more decision
making agents who are operating in a common state space (environment) with a set of actions
available to them. Each action gives a different expected reward in each state for the agents.
A single-agent Markov game is called a Markov Decision Process, where the agent interacts
with its environment represented as a probabilistic transition function. This means that the
environment is fixed in its behaviour. A Markov game is a generalisation of a Markov Decision
Process which allows for multiple adaptive agents with interacting or competing goals. For all
agents in a Markov game, the goal is to maximise some kind of future reward. This section
describes a two-player zero-sum Markov game and its components, which is later used as a
framework to model a football match.
16
3.2 Markov Games
3.2.1
Definition
Following Littman (2001), a two-player Markov game is defined by a tuple
hS, A, O, T, R, i, where
• S = {s1 , s2 , . . . , sn } is a finite set of game states
• A = {a1 , a2 , . . . , am } and O = {o1 , o2 , . . . , ol } are finite sets of actions available to the
agents (A is the set of actions for agent 1 and O is the set of actions for agent 2, the
opponent of agent 1)
• T : S ⇥ A ⇥ O ! ⇧(S) is the transition function. It gives, for each state and one action
from each agent, a probability distribution over states. T (s, a, o, s0 ) is the probability of
ending in state s0 , given that the agents starts in state s, agent 1 takes action a 2 A and
agent 2 takes action o 2 O
• Ri : S ⇥ A ⇥ O ! R for i = 1, 2 is the reward function, giving the expected immediate
reward gained by agent i for taking each action in each state. Ri (s, a, o) is the expected
reward to agent i in state s when agent 1 chooses a 2 A and agent 2 chooses o 2 O
• 0  < 1 is a discount factor. This describes how the relationship between near-term
versus long-term reward is valued. A reward Ri to agent i received j steps in the future is
worth j Rji to the agent now. If = 1, the Markov game is undiscounted
The agents seek the best actions in each state
a way to maximise some measure
P in such
j i
of their long term expected reward received, E{ 1
R
j }. The special case where the two
j=0
agents have diametrically opposite goals allows using only one reward function. This is called
a zero-sum Markov game, in which one of the agents is trying to maximise the reward and the
other, called the opponent, is trying to minimise.
A technique for estimating values for the states in a Markov game is by using reinforcement
algorithms, such as value iteration, to learn a Q-function for each state s and state-action pairs
(s, a, o), in the Markov game state domain S.
3.2.2
Policies and Value Functions
A policy is a description of the behaviour of an agent. It specifies a probability distribution over
actions to be taken for each state, and is denoted ⇡i : S ! ⇧(Ai ). A policy ⇡i , where i 2 {1, 2}
are the two agents, maps a state s 2 S to a probability distribution ⇡i (s, a, o) from Ai . The
probability assigned to actions (a, o) in state s is ⇡i (s, a, o). How good a policy is for an agent
can be evaluated by computing the long term value the agent can expect by following the policy.
The expected future discounted reward for taking actions a for agent 1 and o for agent 2 in state
s and continuing according to policy ⇡a and ⇡o thereafter, is denoted Q⇡i (s, a, o) for agent i.
This can be defined by a set of simultaneous linear equations, one for each state s and agent i,
Q⇡i (s, a, o) = Ri (s, a, o) +
X
X
s0 2S
T (s, a, o, s0 )
⇡a (s0 , a0 )⇡o (s0 , o0 )Q⇡i (s0 , a0 , o0 ).
(3.5)
a0 2A o0 2O
17
Chapter 3. Basic Theory
The function Q⇡i is called the Q-function for the policy ⇡i . Given an initial state s, the agents
should execute policies ⇡a and ⇡o that maximises
X
⇡a (s, a)⇡o (s, o)Q⇡i (s, a, o).
a,o
In most Markov games, the objective is to obtain or learn the optimal policy for the agents
acting in the game, in which case the value of the Q-function is said to be learnt off-policy.
Contrary, if the Q-function is learnt on-policy, it finds the value of the policies being carried
out by the agents in the game. This is the main focus in this thesis, because the policies of
the agents are already known from the data set. More specifically, when the Markov game is
zero-sum and only one agent is active in each state (the other choose no action each time), the
game can be considered a Markov Decision Process. Thus, the mathematics and expressions
can be simplified.
3.2.3
Value of a Policy
Following Poole and Mackworth (2010), consider a Markov Decision Process with a known
stationary policy ⇡. A policy is said to be stationary when it assigns a known distribution of
actions to each state and does not change with time. If a reward criteria is defined for the Markov
Decision Process, the policy has an expected value for every state. Let V ⇡ (s) be the expected
value of following ⇡ in state s. V ⇡ (s) denotes the reward the agent can expect to receive from
following the policy in that given state, and it is defined as the discounted value of all future
rewards with discount factor ,
V1 = E{
n
X
j=0
j
Rj } = R0 + R 1 +
= R0 + (R1 + R2 + · · · +
2
n 1
R2 + · · · +
n
Rn
Rn ) = R0 + V 1 .
Furthermore, let Q⇡ (s, a) denote the expected value of performing action a in state s, and
following policy ⇡ thereafter. V ⇡ and Q⇡ can be defined recursively in terms of each other.
Let R(s, a, s0 ) be the immediate reward of performing action a, leading to a transition to state
s0 when being in state s. An agent who performs actions a in state s can therefore be said to
receive the immediate reward R(s, a, s0 ) plus the discounted expected value of state s0 , V (s0 ).
When the expected value averaged over the possible resulting states is used for Q(s, a), the
following expression is obtained,
⇣
⌘
X
Q⇡ (s, a) =
T (s, a, s0 ) R(s, a, s0 ) + V ⇡ (s0 ) .
(3.6)
s0 2S
An expression for V ⇡ , is obtained by doing the actions specified by ⇡ as
X
V ⇡ (s) = Q⇡ (s, ⇡(s)) =
⇡(s, a)Q⇡ (s, a).
a2⇡(s)
18
(3.7)
Chapter
4
Experimental Setup
This chapter describes the experimental setup in this thesis. First, the data used to build the
models is explained. Next, three different models based on theory from Chapter 3 are presented.
The first is an expected goals (xG) model, developed by the use binary logistic regression in
order to assign values to shots attempted in football. The remaining two models are variations
of Markov game models applied to football, which share many of the same features, but are
nonetheless clearly separated in order to avoid any confusion. Results from running the models
are presented and discussed in Chapter 5.
4.1
Data
The data used in this thesis was collected by Opta (2016), and consists of almost two and a
half seasons worth of matches played in Tippeligaen in the F24 football feed format. Opta
collects the same type of data from every match in 30 leagues and tournaments worldwide.
This type of data is mainly used by businesses and stakeholders in football, such as betting
companies and statisticians working at football clubs. Sports analytics firms, newspapers and
broadcasters are also potential users of such data. The F24 football feed has been used to some
extent in academic work as well, such as in Lucey et al. (2013b) and Kerr (2015). The data is
not publicly available, and one season costs several thousand GBP.
Opta collects the data by the use of human annotators. All ball events that occur around
the ball during a match are stored chronologically with information on team, player, x- and
y-coordinates, outcomes and time stamp. A modified example of the structure of the F24 feed
is shown in Figure 4.1. Values can not be shown due to a non disclosure agreement between the
authors and Opta. F24 files for every match played in Tippeligaen 2014 and 2015 are the main
source of data used in this thesis. With 16 teams competing in the league and 30 match days,
240 matches are played each season. In addition, the matches from the first 13 match days of
the 2016 season are used to test the models out of sample. 69 different event types exist, where
each event can have multiple additional qualifiers associated with it, depending on the type.
For example, a pass has qualifiers like end coordinates, distance, angle and several more, and a
shot has qualifiers like which body part was used to shoot, what type of play it came from and
information on where the shot ended. The x- and y-coordinates are normalised to 100 ⇥ 100 in
the data.
Such a comprehensive list of qualifiers in addition to the list of 69 different event types
results in a complex event space. This lead to a quite extensive job for the authors to get familiar
19
Chapter 4. Experimental Setup
Figure 4.1: Data structure of F24 feed
with the data set. A substantial amount of time has been spent watching video tapes comparing
the data to the tape, in order to fully understand how Opta characterises the different events.
Furthermore, not all the event types in the data set are considered relevant for an approach such
as in this thesis. The data set, which is on the form as one XML file for each match, was read
in C++ using Xcode for Mac, where objects are created for each match and each event.
In order to make the data applicable for the desired approaches, a restructuring of the data
set is carried out. Events such as period start, period end, expected or unexpected player or
referee changes and other events independent of the ball, are removed. Moreover, each match
is divided into sequences, where a sequence stops when the ball goes out of play for a goal kick
or a throw-in, if a goal is scored or if a period is over. For the events in these sequences, some
extra features are calculated and added. These features are described further in the forthcoming
sections. This lead to a data structure as shown in Table 4.7 for the Markov models. The data
used for the expected goals model, is a list of all the shots and goals in the data set, excluding
the own goals.
The data only follows the ball and does not contain information on the position of players
not acting on the ball, and is thus not spatiotemporal. In addition, no event for carrying the ball
across the field is defined by Opta. Because of this, it is possible for apparent jumps between
areas on the field where subsequent events are taking place. However, by looking at the previous
and current event, inferences can be made about the position of the ball between the events as
in Lucey et al. (2013b).
Data from all the matches are collected by hand and are verified at least once after each
match. Mistakes that are revealed are corrected or removed continuously, and the data can
be considered clean, complete and accurate for all practical purposes. In addition, the aforementioned comparison of video and data by the authors of this thesis did not reveal significant
discrepancies between the hand annotated data and what actually occurred. Table 4.1 shows an
overview of some simple descriptive statistics for the data set.
Table 4.1: Simple descriptive statistics for the dataset
Tippeligaen
Matches
Goals*
Number of shots*
Number of passes
*
20
2014
2015
2016
240
240
104
735
774
261
6,894
6,598
2,763
199,675 198,168 88,277
including own goals, 27 in 2014 and 22 in 2015.
4.1 Data
From Table 4.1 it can be seen that a pass is the most frequent type of event in the data set.
The number of events vary from match to match, depending on the playing style of the teams
and other factors. Furthermore, it can be seen that the numbers seem fairly consistent across
seasons.
Figures 4.2 and 4.3 illustrate simple examples of what can be calculated from this kind of
data.
(a) All passes start
(b) Passing accuracy start
(c) All passes end
(d) Passing accuracy end
Figure 4.2: Illustration of passing data in terms of pass counts and passing accuracies
Figure 4.2 shows four maps of the field with pass counts to the left and passing accuracies
to the right. The data is always from the perspective of the attacking team playing from left to
right. As one should expect, most passes are hit in the middle of the field, while the passing
accuracy is higher on the defensive half.
Figure 4.3 consists of four maps over the attacking half of the field. The thick black lines in
the maps illustrate the 18-yard box. Again a couple of unsurprising observations can be made.
The shooting accuracies are higher closer to the goal, and the accuracy decreases if the shot was
a header or if the shot was assisted by a cross.
21
Chapter 4. Experimental Setup
(a) Regular shots
(b) Headed shots
(c) Free kicks
(d) Shots from crosses
Figure 4.3: Illustration of shot data in terms of conversion rates
4.2
Expected Goals Model
The method of binary logistic regression, introduced in Section 3.1, is used to develop an xG
model in order to rate primarily offensive players in Tippeligaen 2014 and 2015 based on their
efficiency in converting shots into goals. This section describes the model building approach and
the explanatory variables that are tested in order to obtain an xG model for assigning likelihoods
of scoring on shots attempted. This model is denoted as the xG Model hereafter.
4.2.1
Model Building Approach
Three distinctive types of shots are considered in the xG Model. They are free kicks, penalty
kicks and regular shots (all shots other than free kicks and penalty kicks). This distinction is
chosen due to the different nature of these situations. Free kicks and penalty kicks are static
events where the shooters are completely undisturbed by defenders, contrary to regular play
where more factors are influencing the outcome of a shot. The general idea is still the same for
all three types, namely to assign a likelihood for scoring on each shot.
The outcomes of 12,781 regular shots and 558 free kicks from Tippeligaen 2014 and 2015
are used as the binary dependent variable (1 for goal, 0 for miss) in two separate logistic regression models. All penalty kicks are taken from the same position and therefore have almost
no variation to account for. Thus, the likelihood of scoring on a penalty is estimated by the
conversion rate observed from the 101 penalty kicks taken over the two seasons. An example
of the structure of the shot data is shown in Table 4.2.
Three free kicks that ended in a goal are taken out of the data set. The three goals are Stian
Ringstads 1-0 goal for Lillestrøm against Odd in 2014, goalkeeper Håkon Opdals 1-1 goal for
Start against Vålerenga in 2015 and Espen Børufsens 3-1 goal for Start in the game against
Sarpsborg 08 in 2015. They are considered outliers because the players that took the free kick
did not intend to score, and a substantial amount of luck was involved for each of them to end
up in a goal.
A large set of explanatory variables are used as a starting point for the binary logistic regressions for regular shots and shots from free kicks. They are further discussed below. The method
22
4.2 Expected Goals Model
Table 4.2: xG Model: Structure of shot data
Var1
21.1
5.1
18.0
..
.
11.4
Var2
Var3
0.1245
1
0.2458
1
0.0459
0
..
..
.
.
0.1648
0
Free kick
0
1
1
..
.
...
...
...
...
..
.
Varn
1
0
0
..
.
Goal
1
0
0
..
.
0
...
1
1
of backward elimination described in Groebner et al. (2011) is used to determine a set of statistically significant explanatory variables for the two regression models for regular shots and free
kicks. Regression coefficients and p-values are estimated using statistical software SSPS which
use the method of maximum likelihood.
4.2.2
Explanatory Variables
As mentioned above, a large set of explanatory variables are considered for the two regression
models for regular shots and free kicks. They are shown in Table 4.3 along with the type of
variable and whether it was calculated or determined by the authors or not.
Figure 4.4: xG Model: Illustration of angle and length variables for a shot. Red lines show the length
and orange arrows show the angle. Angle measured in radians
The distance to goal and angle to goal (measured in radians) are continuous variables and
are illustrated in Figure 4.4. It is believed that a longer distance to goal decrease the likelihood
of scoring, because the shooter has to aim more precisely and the goalkeeper has longer time
to react. The angle describes how much the shooter can see of the goal, and it is believed that
a higher angle is favourable for the outcome. Current match status is the goal difference in the
game from the perspective of the player that attempted the shot, and the home or away variable
is whether the player played at home or not.
23
Chapter 4. Experimental Setup
Table 4.3: xG Model: Variables considered in binary logistic regression
Variable description
Distance to goal
Angle on goal
Current match status
Home- or away game
Difference in team strength
Assisted by pass
Assisted by rebound
Assisted by long ball
Assisted by cross
Assisted by throughball
Assisted by head pass
Assisted by shot
Type of play: Regular play
Type of play: Set piece
Type of play: Corner
Type of play: Throw-in
Type of play: Fastbreak
Number of preceding take on
Finished by a header
Finished by other body part
Variable type
Continuous
Continuous
Categorical
Binary
Nominal
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Binary
Nominal
Binary
Binary
Calculated
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
Yes
No
No
Difference in team strength is assessed using the Elo rating system, described in Hvattum
and Arntzen (2010). For the player attempting the shot, the difference in Elo rating between
the two teams playing in a match is used on each shot. Players on high quality team are likely
better shooters than on lower quality teams. However, the lower quality teams are likely to
defend more, which can decrease the likelihood of scoring due to increased defender proximity.
Seven binary variables describe how the shot was assisted, that is how the ball was brought
into the shooting position. They are all determined by looking at the event preceding the shot.
They are believed to influence the likelihood of scoring differently. For example, if a shot was
assisted by a cross the likelihood is believed to decrease because it makes the job of the shooter
more difficult, which is supported by Figure 4.3(d). The opposite can be said regarding a shot
following a rebound, where it often is a clear chance of goal either from a save by the goalkeeper
or a rebound off the post.
Five binary variables describe the type of play the shot came from. They are defined by
the human annotators in Opta and are present as a qualifier for each shot in the data set. Such
definitions make them subject for human bias. Especially the qualifier for fastbreak is vaguely
defined and seldom occur in the data.
The number of preceding dribbles is the number of successful dribbles a player did before
shooting. Most likely, the player attempted to move the ball to a better area by dribbling off defenders before a shot, and is therefore believed to influence the likelihood of scoring positively.
The two binary variables in the bottom of the table describe what body part the player
making the shot used to finish. They are qualifiers associated to every shot by Opta, and are
both believed to negatively influence the likelihood of scoring.
24
4.3 Markov Game Model 1
4.2.3
Application of the Expected Goals Model
After the method of backward elimination is applied for regular shots and free kicks separately,
one or more statistically significant variables remain. They constitute the xG Model for the
two types of shots, and their coefficient values are used as input in Equation (3.4) to assign
a likelihood for converting regular shots or free kicks into goals. As mentioned earlier, the
likelihood of scoring on a penalty kick is estimated by using the observed conversion rate.
These likelihoods are used to rate players in terms of efficiency by looking at how many goals
they actually scored, compared to how many they were expected to score by aggregating the xG
value from the shots they attempted. The efficiency measure used to rank players from the xG
Model is defined as actual goals divided by expected goals and is denoted G/xG throughout this
thesis.
It is also possible to aggregate the xG values for each team, both in terms of xG for and xG
against by looking at the shots a team attempted and received, respectively. From this, a league
table based on the difference in xG (xG for minus xG against) can be generated.
4.3
Markov Game Model 1
A Markov game, formally introduced in Section 3.2, is used to develop a model for evaluating
all player involvements, not only shots. Football matches are modelled as a Markov game where
the two teams playing represent the agents in the game. The two teams have diametrically
opposite goals, which makes it a zero-sum Markov game. In addition, the value of the state and
state-action pairs are learnt on policy. As described in Section 3.2.2, these specifications make
it possible to consider the game as a Markov Decision Process, which simplify the mathematics.
The different parts of this model are described in the following subsections, and the model is
referred to as Model 1 in the rest of this thesis.
4.3.1
State Space: Context Variables and Field Zones
The literature review on sports analytics in Section 2.1 suggested that the context variables
match status, match location and manpower difference on the field influence the dynamics of
the game. These three context variables forms the basis for the state space in this model. Early
Markov process models for football, such as Hirotsu and Wright (2002, 2003a), used a simple
four-state model with possession and goals constituting the state space. This state space include
two of the most fundamental attributes of football, but is still a rough approximation of a highly
complex game. In Model 1, the state space is expanded to include context variables and the
position on the field where an action is taken.
The context variables provide information on what phase or condition the game is in. The
general idea of introducing such variables is that they influence how players on a team make
decisions. When a team is leading, they are more likely to defend instead of continuing launching forward to extend their lead. Similarly, if a team gets a player sent off they are more likely
to sit back and attempt to launch counter attacks in order to score. Context variables used in the
state space are listed in Table 4.4 along with their respective values.
The number of time periods, T P , in a match is chosen to four. This is done to capture the
effects the time left to play has on a match. When a game enters its ending phase, scoring a goal
can possibly be more valuable than scoring earlier in the match because the opposition have less
time to respond. Given the four time periods, T P changes value at 23 and 68 minutes played as
25
Chapter 4. Experimental Setup
Table 4.4: Model 1: Context variables
Variable
Notation Values
Time period
TP
[1, 2, 3, 4]
Match status
MS
[ 1, 0, 1]
Manpower difference M D
[ 2, 1, 0, 1]
well as at half time. Because of overtime, T P = 2 and T P = 4 can be slightly longer than the
other two. Match status, M S, determines which team is leading a match. If the home (away)
team is leading, M D = 1 ( 1) and M S = 0 if the teams are drawing. Manpower difference,
M D, is the difference in players on the field between the two teams. M D = 1 ( 1) if the away
(home) team have gotten a player sent off, and M D = 0 when they have an equal number of
players on the field.
Any combination of the three context variables constitute a pure context state. There are
4 · 3 · 4 = 48 unique context states, which is derived from multiplying the number of possible
values for the three variables. 34 of the possible 48 context states occur at least once in the data
set.
It is possible to add more context variables to make the model more detailed. Examples
of other context variables that can affect the state of a game are the relative strength between
the teams playing a match and the yellow card difference. Introducing more context variables
would enlarge the state space, and potentially make the model more realistic. However, by
extending the state space there is a chance it could make the number of occurrences of some of
the states too low. With data from no more than 480 matches, only the three situational variables
considered to have the largest impact on the game are included.
Instead, the state space is extended to include location on the playing field. The football
field is split into 15 zones as shown in Figure 4.5, where the attacking team is playing from left
to right. If the attacking team lose possession in zone 15, the defending team wins it in their
zone 1. As seen from the figure, the grid is not symmetrical about the centre line, but is rougher
in the defensive half than the offensive. The difference in value between having the ball in zone
1 by the sidelines or in the centre of the field is likely very small. This can not be said about the
difference between zone 14 and 15 in the offensive half where the goal is situated.
Figure 4.5: Model 1: Field zones
The grid in Figure 4.5 can be changed to become symmetrical and/or include a higher number of zones. It is chosen like this to avoid a too large state space. With the amount of data
available, a larger state space could give states with too few observations to build the model
from.
In addition to the context variables and the zone of the field, a state also contain information
26
4.3 Markov Game Model 1
on what team is in possession of the ball. This is done in order to keep track of who is executing
the action. Figure 4.6 is an illustration of the information that is included in a state in Model 1.
There are 2 · 4 · 3 · 4 · 15 = 1440 possible states in the state space. Of these, 902 occur at least
once in the data set.
Figure 4.6: Model 1: Definition of a state
4.3.2
State Transitions: Actions
A central part of any Markov game is the transition between states. For each state in the state
space, there is a transition function giving probabilities of ending up in a state given one action
from each agent in the game. In Model 1, a state transition occurs when the two agents make
actions to move the ball to another area of the field (a change of zone) or they make an action
that change the context variables (a red card or a goal occurs). The actions available to the team
in possession is denoted A and are shown in Table 4.5. For the defending team, the action space
O, shown in Table 4.6 is more limited. In both the sets, the no action is an alternative which
is used by the defending team most of the time. For each state transition, one team always
performs no action, which make only one agent active. Hence, this can be interpreted as a
Markov Decision Process. This reflect how football is played, and is also in accordance with
the data set which does not include positional data on players other than the ones acting on the
ball. A change of time period occurs automatically independent of the actions.
All actions in Table 4.5 and 4.6 are defined by Opta in the data set, except the Ball carry
action which is made by the authors. It is done to avoid jumps from one state to another without
an action being made. An example of this is if a player receives a pass in zone 9 and run with the
ball to zone 13 and takes a shot. Between the end coordinates of the pass and the shot location,
there might be an apparent jump between states without an action. The Ball carry event prevents
this from occurring. Caution has been taken when introducing the Ball carry action, and it is
only included when there is no doubt that a Ball carry actually occurred.
Table 4.5: Model 1: A, action set for the team in
Table 4.6: Model 1: O, action set for the team not
possession of the ball
in possession of the ball
Pass
Take on
Shot
Aerial duel
Chance missed Cross
Long pass
Throw-in
Free kick taken
Ball carry
Corner won
No action
Dispossessed Yellow/red card
Foul
Ball recovery
Ball touch
Tackle
Clearance
Yellow/red card
Interception
Aerial duel
No action
The state transitions are calculated from observed play. Table 4.7 shows an example of
how the data used in this model is structured. The data consists of a comprehensive list of
27
Chapter 4. Experimental Setup
events, which is iterated through in C++ in order to first create a vector of the occurring states.
The occurrences of each state are then counted and denoted Occ(s), as well as the number of
transitions from s to s0 given actions a 2 A and o 2 O from the two teams, Occ(s, a, o, s0 ). It is
possible for a team to have successive actions without the opposing team interfering. The state
transition probabilities are estimated by
T (s, a, o, s0 ) =
Occ(s, a, o, s0 )
.
Occ(s)
Table 4.7: Model 1: Structure of data
Match
1
1
1
1
..
.
Sequence
1
1
1
2
..
.
Player
21544
56897
26423
13469
..
.
TP
1
1
1
1
..
.
MS
0
0
0
1
..
.
MD
0
0
0
0
..
.
Zone
3
2
1
11
..
.
Team
Home
Home
Away
Home
..
.
Action
Pass
Long pass
Aerial duel
Throw-in
..
.
Figure 4.7 shows an example of a play sequence used to build the state space, and how the
state transitions occur. It is taken from the match Tromsø - Sarpsborg 08 in 2015. As can be
seen from the figure, the away team takes a throw-in from zone 12 which is intercepted by the
home team in their zone 2. From here, a take on from the home team takes place, followed
by two passes and then a shot from zone 14. The shot can go to one of the two artificial shot
outcome states, depending on whether it was a goal or not. In this particular example, the shot
did not end up in a goal.
Figure 4.7: Model 1: Illustration of states and transitions. Example from Tromsø - Sarpsborg 08 in
2015
28
4.3 Markov Game Model 1
4.3.3
Reward Function and Value Iteration Algorithm
The reward represents the objective of the agents in a Markov game. The ultimate objective in
football is to score more goals than your opponent. The objective of your opponent is the direct
opposite of yours. Motivated by this, the states in the state space with a goal are given reward,
R(goal) = 1 if the home team scores, and R(goal) = 1 if the away team scores the goal. This
way of rewarding states favour the offensive players on the field because they are playing closer
to the opposition goal and therefore more often comes in a scoring position. Regardless of
that, the defensive players with many ball recoveries and duels won are also rewarded, but most
likely with smaller values. It is possible to define rewards to other states as well, depending on
what one wants to measure, but the obvious choice for football is goals.
In order to assign a value to each state-action pair, a Q-function which updates the values
in each iteration can be used. Since the objective is to find the values of a known policy, let
Equation (3.6) from Section 3.2 be the starting point to define a Q-function
⇣
⌘
X
Q⇡ (s, a) =
T (s, a, s0 ) R(s, a, s0 ) + V ⇡ (s0 ) ,
(3.6)
s0 2S
while Equation (3.7) is the starting point when defining an expression for the value of a state
X
V ⇡ (s) = Q⇡ (s, ⇡(s)) =
⇡(s, a)Q⇡ (s, a).
(3.7)
a2⇡(s)
The policy for a state in this model, ⇡(s), can be interpreted as the probability of performing
the different actions a in state s, which for each state and each action is denoted by
⇡(s, a) =
Occ(s, a)
.
Occ(s)
From this, the expected value of being in state s0 for the home team can be written as
1
VH (s ) =
·
Occ(s0 )
0
X ⇣
Occ(s
0
, a0H )
a0H 2A[O
· QH (s
0
, a0H )
⌘
(4.1)
.
Furthermore, the probability of ending in state s0 when performing action a, is denoted
Occ(s, a, s0 )
T (s, a, s ) =
.
Occ(s, a)
0
Hence, Equation (3.6) can be written as the following when the home team (H) is performing
action a
QH (s, aH ) =
X⇣
1
Occ(s, aH , s0 ) · RH (s, aH , s0 ) +
Occ(s, aH ) s0 2S
· VH (s0 )
⌘
.
(4.2)
Equation (4.2) is the Q-function which is applied in the value iteration algorithm in order
to learn the values of the state-action pairs in Model 1. The values of each state, VH (s0 ), are
calculated by Equation (4.1).
The input to the value iteration algorithm is the vector containing all the states, including
information of number of occurrences and state transitions. The algorithm shown in Algorithm
29
Chapter 4. Experimental Setup
1 is run in C++ using Xcode for Mac, and runs until convergence or until the maximum number
of iterations is reached. A relative convergence criteria of 0.0001 and a maximum number of
iterations of 10,000 is used. The use of relative convergence implies that the algorithm will terminate only when the Q-values for the state-action pairs are being increased by a small amount.
Conversely, the algorithm will continue when the Q-values increases with larger amounts.
Require: Markov game model, convergence criterion c, maximum number of iterations M
1: lastV alue = 0
2: currentV alue = 0
3: converged = f alse
4: for i = 1; i  M ; i
i + 1 do
5:
for all state-action pairs (s, a) in the Markov game model do
6:
if converged == f alse P
then
1
7:
Qi+1 (s, a) = Occ(s,a) (s0 )2S (Occ(s, a, s0 ) · (Ri (s, a, s0 ) + Vi (s0 )))
8:
currentV alue = currentV alue + |Qi+1 (s, a)|
9:
end if
10:
end for
11:
if converged == f alse then
alue lastV alue
12:
if currentV
< c then
currentV alue
13:
converged = true
14:
end if
15:
end if
16:
lastV alue = currentV alue
17:
currentV alue = 0
18: end for
Algorithm 1: Dynamic Programming for Value Iteration
The Q-function is not discounted ( = 1), implying that the players in the Markov game
are indifferent whether a goal is scored after a long or a short sequence of actions. It is possible
to argue that goals scored after fewer moves could be more worth than long build-ups ending in
a goal. However, goals are scarce in football and it is not an unreasonable assumption that all
goals should be valued the same regardless of how they came about.
Again, it is worth mentioning that the goal of Model 1 is not to find the best possible policy
or strategy for the agents in each state, as in many other Markov games. This is due to the
known transition probabilities, which are estimated from what is observed in the data. Instead,
the goal is to find the value of the policy that is being carried out by the agents. However, it is
worth mentioning that the optimal action to perform in each state, can be calculated based on
the immediate reward generated by the action. This can be done by the formula
⇤
⇡H
(s) = arg max Q(s, aH ).
(4.3)
aH 2A[O
Since the purpose of the model is to assign a value of a given policy, this action can not be
considered an optimal policy. Instead it can be referred to as an optimal action in a state. These
formulae are also applied on the other agent in the Markov game, the away team.
30
4.3 Markov Game Model 1
4.3.4
Valuing Individual Player Actions
After the value iteration algorithm has converged, all states and state-action pairs have been
assigned a value. On the basis of this, each player involvement can be assigned a value based
on its impact on the game. Several functions for the impact can be used. For this model,
especially two impact-functions are considered. They are
IH (s, aH , aA , s0 ) = VH (s0 )
QH (s, a)
(4.4)
IH (s, a, s0 , a0 ) = QH (s0 , a0 )
QH (s, a).
(4.5)
and
With the former, Equation (4.4), it can be said that the performing player receives the difference between the value of state s0 the action ended in and the value of performing action a
in state s. With the latter, Equation (4.5), players receive the difference between the value of
performing action a0 in state s0 and the value of performing action a in state s. With both these
formulae, the model captures the value of how the action affected the state the match was in.
The latter accounts for what the given action actually lead to with the average impact of this
action in the given state, while the former compares the value of performing the action with
the average value of being in state s0 . Based on this, the latter is chosen as the impact function
in order to capture the effect of what the action actually resulted in. An example and further
discussion of this effect are given below. Positive impacts are in favour of the home team while
negative impacts are in favour of the away team. As mentioned above, the output from the
value iteration algorithm are values for each state and state-action pairs. Figure 4.8 shows an
illustrative example of state values for both the home and away teams in the simplest context
available: T P = 1, M D = 0 and GD = 0.
(a) Home team zone values
(b) Away team zone values
Figure 4.8: Model 1: Zone values of state space for illustrative purposes.
T P = 1, M D = 0 and GD = 0
31
Chapter 4. Experimental Setup
From Figure 4.8(a) it can be seen that the state in the bottom right corner for the home team
(zone 15) has the value VH (15) = 0.0175 and the adjacent state in front of the goal (zone 14)
has the value VH (14) = 0.0635. For simplicity, assume that a cross from zone 15 in this context
has the same value as the state, QH (15, Cross) = 0.0175 and that a shot from zone 14 in this
context has the value Q(14, Shot) = 0.1250. If a player on the home team crosses the ball from
zone 15 to zone 14 in this context and the following action was a shot from zone 14, he would
gain an impact of
IH = QH (14, Shot)
QH (15, Cross) = 0.1250
0.0175 = 0.1075.
On the other hand, assume that a pass from zone 14 has a value of Q(14, P ass) = 0.05. A
cross from zone 15 that lead to a pass in zone 14 would then only receive an impact of
IH = QH (14, P ass)
QH (15, Cross) = 0.050
0.0175 = 0.0325.
From this it can be seen that the impact value of an action can be highly dependent on the
next action when using Equation (4.5). If Equation (4.4) is used as the impact function, the
player would have received an impact of
IH = VH (14)
QH (15, Cross) = 0.0635
0.0175 = 0.0460,
independent of which type of action is performed next.
From this example it is possible to argue that the Equation (4.5) captures how the player
involvement affected the game in a better way than Equation (4.4). From another point of view,
it might be unfair that a player is punished if his teammate, which he passed the ball to, loses
the ball. In the long run, if a player performs the same pass (a pass between the same two zones)
multiple times, his total impact should converge to a value that is righteous for the player. If
a given player is considered a good passer, it is more likely that a teammate that receives a
pass might be in a better position to do something useful with the ball, compared to if the pass
was from a player that is considered a worse passer. Based on this, it is believed that the latter
impact function is fair over the course of a whole season. However, this impact function leads to
a suspicion that the model might favor players on the better teams because of better teammates.
Values from the impact function in Equation (4.5) can be aggregated for each player in every
game over the course of a season. This enables an evaluation of a players total performance
over a season. In addition, team performances can also be evaluated. In each game, the values
for both the home and away team can be aggregated and compared to determine which team
performed the best. For instance, if the home teams impact totalled 3.5 and the away teams 2.5,
the home team receives a value of 1 and the away team a value of -1 for that match. Again,
these values can be aggregated over a season, from which a league table can be created.
4.4
Markov Game Model 2
As mentioned in the introduction to this chapter, two Markov game models are developed for
evaluating all player involvements, not only shots. This section describes the second Markov
game model, hereby referred to as Model 2. It is built upon the basic principles from Section
3.2. The same specifications as for Model 1, which make it possible to consider it as a Markov
Decision Process, apply. The main difference between the two models is the amount of information included in a state, which now includes what action was made. With this definition,
32
4.4 Markov Game Model 2
after the states have been assigned a value, the value of the state actually represents the value
of that specific action. An important drawback with this model is that the role of the agents
choosing an action in each state becomes less apparent, making the state transitions somewhat
more abstract. The different parts of Model 2 are described in the following subsections.
4.4.1
State Space: Context Variables and Field Zones
Following Section 4.3.1, it can be said that context variables such as match status, match location and manpower difference influence how the players on the field make their decisions. The
context variables used in Model 2 are the same ones as in Model 1 and are repeated in Table
4.8.
Table 4.8: Model 2: Context variables
Variable
Notation Range
Time period
TP
[1, 2, 3, 4]
Match status
MS
[ 1, 0, 1]
Manpower difference M D
[ 2, 1, 0, 1]
The next similarity with Model 1 is the extension of the state space to include the location
on the field, repeated in Figure 4.9. As seen earlier, the field is divided in 15 zones which in
combination with the context variables leads to a state space of potentially 1440 states.
Figure 4.9: Model 2: Field zones
4.4.2
State Space: Actions
The actions available to the two agents are the same as for Model 1. In Model 1, the effect of
the different actions is captured through the impact function. For Model 2, the type of action
is included in the state definition, in order to separate the different types. In addition, the zone
which an action ended in is also included in the state definition. This is done in order to capture
the difference between especially passing events, for instance a pass from zone 2 to zone 3 and
a pass from zone 2 to zone 5. As previously mentioned, Opta delivers the end-coordinates for
passing events making this possible.
Another feature which might separate an action from another, is whether it was successful
or not. Opta captures this through their outcome parameter. Table 4.9 shows the definitions
of the two outcomes of each type of action. In order to separate a successful action from an
unsuccessful one, the outcome of each action is also included in the definition of a state. Figure
33
Chapter 4. Experimental Setup
4.10 shows a summary of what kind of information that is included in the definition of a state
in Model 2.
Table 4.9: Model 2: Overview of the outcome parameter for the actions
Action
Pass
Take on
Foul
Corner won
Tackle
Interception
Clearance
Shot
Yellow card
Aerial duel
Ball recovery
Dispossessed
Chance missed
Ball touch
Cross
Long pass
Throw-in
Free kick taken
Ball carry
Outcome = 1
Successful
Successful
Foul won
Always set to 1
Wins possession
Always set to 1
Always set to 1
Always set to 1
Always set to 1
Duel won
Always set to 1
Always set to 1
Ball simply hit the player
Successful
Successful
Successful
Successful
Always set to 1
Outcome = 0
Unsuccessful
Unsuccessful
Foul conceded
The other team retains possession of the ball
Duel lost
Always set to 0
Unsuccessful control of the ball, possession lost
Unsuccessful
Unsuccessful
Unsuccessful
Unsuccessful
-
The inclusion of all these features lead to a very large and complex state space. Since the
ball can go out of play or into one of the goals, an action can end in 17 different zones. 19
different actions can be carried out, with outcome equal to 0 or 1. This, in combination with
the context variables, yields a number of 930,240 possible states, but the majority of them are
infeasible due to the nature of the action being carried out. For instance, it is impossible to hit
a cross from zone 3 to 5. In the data set, only 34,697 states occur at least once. These numbers
are indeed very high compared to the number of events in the data set, but the main idea behind
this model is that when a state has been assigned a value, this value is actually the value of that
specific involvement.
Figure 4.10: Model 2: Definition of a state
34
4.4 Markov Game Model 2
4.4.3
State Transitions
The state transitions become less intuitive in Model 2 because almost all information is stored
in the states. Moreover, with the action type and in which zone the action ended stored in the
definition of a state, the decision of which action to perform for the agents is not visible at
first sight. The structure of the data used in Model 2 is the same as for Model 1 (see Figure
4.7), and again the comprehensive list of events is iterated through in C++ in order to create the
state space. The occurrences of each state are counted and denoted Occ(s), in addition to the
number of transitions between state s and s’, Occ(s, s0 ). From this, the transition probabilities
are calculated by
Occ(s, s0 )
T (s, s0 ) =
.
Occ(s)
The rationale behind state transitions like this, is more in terms of what the performed action
lead to, as opposed to the change of context or state because of an action as for Model 1. With
this is mind, the role of the agents in Model 2 vanishes to some extent. When a player has
the ball in a given zone, the action he performs, to which zone and the outcome of the action,
decides which state the game is in. The following state is decided by the subsequent action.
Hence, the role of the players as agents is not as visible as for Model 1. Figure 4.11 illustrate
an example sequence modelled as described above, and is the same sequence used for Model 1
in Figure 4.7.
Figure 4.11: Model 2: Illustration of states and transitions. Example from Tromsø - Sarpsborg 08 in
2015
From Figure 4.11, it should be evident that a state includes all the information except the
transitions. The first state in the figure is an unsuccessful (outcome is equal to 0) throw-in for
the away team from zone 12 to zone 12. Since this throw-in was unsuccessful, the other team
(the home team) has the ball, and therefore the next state is and interception for the home team,
35
Chapter 4. Experimental Setup
which again is followed by a take on. The next actions are passes from zone 8 to 9, and from
zone 9 to 14, followed by a shot from zone 14. As mentioned before, this particular shot did not
end up in a goal.
4.4.4
Reward Functions and Value Iteration Algorithm
The reward function and the value iteration algorithm are the same as for the previous model.
Reward is given for states where a goal occurs, and states are rewarded 1 when the home team
scores and -1 when the away team scores. These rewards are given to the artificial shot outcomeevents, as found in the R(s) values in Figure 4.11. The Q-function used in this model is almost
the same as for the previous one, since this model also is on-policy. As Equation (4.6) shows,
the only difference is the replacement of the values for the state-action pair with Q-values for
the states. This is mainly due to the absence of actions as state transitions. The Q-function for
Markov Game Model 2 is defined as
QH (s) =
X ⇣
1
·
Occ(s, s0 ) · RH (s, s0 ) +
Occ(s)
0
(s,s )2S
· QH (s0 )
⌘
.
(4.6)
This Q-function is also undiscounted ( = 1) of the same reasons as for the previous model.
As for Model 1, the input to the value iteration algorithm is the vector containing all the states,
including information regarding occurrences and state transitions. The algorithm is then run in
C++ using Xcode for Mac.
When again considering Figure 4.11, the unsuccessful throw-in exemplifies how the outcome parameter captures an important feature in football. Since the throw-in is unsuccessful, the following states consist of events performed by the other team, which leads to a nonfavourable contribution to the value of the state. On the other hand, if the throw-in was successful, it should in most cases be followed by an action performed by the same team, which yields
a favourable contribution.
4.4.5
Valuing Individual Player Actions
After the value iteration algorithm has converged, each state has been assigned a value. Once
again, since all the information is included in the states, the value of a state equals the value of
the involvement described by the state properties. This leads to the following function for the
impact of an action, defined simply as
IH (s) = QH (s).
(4.7)
As for Model 1, these values can be aggregated over a match or a whole season for each
team or for each player. An important artifact with an impact function like this, is that the model
prefers offensive involvements to defensive ones, due to them being closer to the objective of
the game. It is also worth mentioning that a shot is valued on the basis of the value of taking
a shot in the given context, Q(s), not whether the shot ended up in a goal or not. Thus, a
player that misses a shot receives the same value as a player that scored on the same shot. The
motivation behind this is to be able to evaluate players on a basis where the outcome from shots
is reduced. Hence, it can be said that the players are being evaluated in terms of their impact on
creating scoring opportunities, where the player receives a high value if the involvement creates
a scoring opportunity that is likely to result in a goal.
36
4.5 Model Evaluation
The output of the model can be utilised to examine which action is the best in a given
context. If one wishes to find the best action in the context of for instance T P = 3, M S =
1, M D = 0, H = 1 and Z = 3, one can compare the different state values which has these
properties. Table 4.10 shows an example of values for a pass in this context for illustrative
purposes.
Table 4.10: Model 2: Pass values for the context T P = 3, M S = 1, M D = 0, H = 1 and Z = 3
Type
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
ZoneEnd
1
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
7
8
9
10
11
Outcome
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
Goal
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
StateId
2856
466
465
6204
562
528
514
7185
25416
17777
9984
17049
1382
1385
6224
6189
539
10235
8440
21362
24731
Occurrences
41
469
1219
198
132
161
186
25
7
14
20
13
153
53
36
43
49
19
16
21
14
Value
0.0013
0.0023
0.0045
0.0056
0.0082
0.0079
0.0065
0.0091
0.0407
0.0074
0.0108
-0.0066
-0.0005
0.0002
0.0005
-0.0002
0.0003
0.0012
0.0023
0.0012
0.0020
Hypothetically, if a pass was the only option for a player being in this context, a successful
pass to zone 9 is optimal. This can be seen in Table 4.10, with a successful pass to zone 9 having
a value of 0.0407, which is the largest value in the rightmost column. Furthermore, it can be
seen from the table that a successful pass is better than the corresponding unsuccessful one. It
is worth noting that zone 12, 13, 14 and 15 are unreachable from zone 3 with a regular pass. A
pass from zone 3 to one of these zones is considered a long pass by Opta
4.5
Model Evaluation
To evaluate the performance of the three models developed in this thesis, their reliability and
validity are investigated individually.
The reliability of the xG Model is assessed by looking at the correlation coefficients between
2014 and 2015 for the performance measures used to rate players. High correlations would
indicate that the model is able to rate players consistently across the seasons, and can give
insight on how reliable the model is for decision making. To make inferences on the validity of
the xG Model, the area under the ROC curve, introduced in Section 3.1, for the two shot models
37
Chapter 4. Experimental Setup
are assessed. A high area under the ROC curve is desirable, as it indicates that the models have
good discrimination abilities.
The reliability of the two Markov game models are assessed by looking at the correlation
between the aggregated impact value of the players per 90 minutes played across the two seasons. In a game like football, players are believed to perform at a relatively stable level across
two seasons, which is the motivation for exploring the correlation between seasons as a measure
of reliability. A player can improve or get worse from one season to another, but the difference
should not be too large. Positive correlation coefficients between the two seasons would indicate that the models are able to assign high positive values to good players in both seasons
and not to players that occasionally did something extraordinarily good. Only players that have
played at least 900 minutes (one third of a season) in both 2014 and 2015 are considered in the
assessment of the reliability. It is believed that the effects of randomness should be limited if a
player has played at least 900 minutes.
Tests of the validity of the two Markov game models are done by looking at correlations
to three benchmarks. No objective player ratings from Tippeligaen are known to the authors,
which makes this part of the model evaluations especially challenging. However, possible indicators of the level of a player might be their market value, salary or subjective player ratings
by journalists and pundits. Some of the largest Norwegian newspapers and TV stations provide
player ratings after each match solely based on subjective opinions. They form the basis for
an average value of a players performance during a season. Two of the best known providers
of such ratings are the broadcaster TV2, who owns the TV rights of Tippeligaen, through their
statistical web page Altomfotball(Altomfotball (2014) and Altomfotball (2015)) and the largest
newspaper in Norway, VG (VG (2014) and VG (2015)). Investigating the correlation coefficients between the ratings of the two Markov game models and the two subjective player
ratings can give insight on whether players that are considered the best by journalists, also are
considered good players by the models. The ratings of the journalists pundits are subjective,
thus observing a positive correlation coefficient between the ratings and the developed models
would be desirable, but is not a goal in itself. The purpose of making such models is to ensure
objective rating of football players free from human bias to measure performance and identify
talent.
In addition, the results from the models are compared to the market value of the players provided by Transfermarkt (2016). For each season, the market value of the players are extracted
from the website of Transfermarkt at year end 2014 and 2015. The values provide a more
objective benchmark for the results from the Markov game models than the journalist ratings.
Correlation coefficients are again the measure of how valid the results are. Positive correlation
coefficients between values from Transfermarkt and the models would indicate that players with
high market value also are the best in the rating from the models. These correlation tests are not
done on the xG Model, since the purpose of the xG Model is to measure the efficiency, G/xG,
of mainly offensive players. The measure of efficiency is not as important for a defender as for
a forward, and hence the values are not directly comparable with the ratings from Altomfotball
and VG, as well as the market values from Transfermarkt.
38
Chapter
5
Evaluating Results
In this chapter, the results from implementing the models described in Chapter 4 are presented
and discussed. The variables included in the xG Model are first presented and interpreted.
Further, the reliability and validity of the xG Model is evaluated, before the results in the form
of a player rating are presented and discussed. For the two Markov game models, the reliability
and validity are discussed before the evaluation of the results, which are presented as top 10
lists for the different player positions. Players that have been sold to foreign clubs, presumably
due to a noticeable performance, are marked with an asterisk in all tables. In the last part of this
chapter, the results are evaluated with regard to the research questions from Section 1.2.
Results from Tippeligaen 2015 are presented in this section, while the results from Tippeligaen 2014 can be found in Appendix B.1, C.1 and D.1 for the xG Model, Model 1 and 2,
respectively.
5.1
Expected Goals Model
As mentioned in Section 4.2, the xG Model consists of three distinctive types of shots: regular
shots, free kicks and penalties. First, the significant explanatory variables for regular shots
and free kicks are presented and interpreted, before the reliability and validity of the two shot
models are assessed. Ratings based on the obtained values from the xG Model are presented
and discussed in the last part of this section.
Players are evaluated on five different performance measures: the number of goals (G), total
xG, the number of shots (S), xG per shot (xG/S) and the number of goals divided by total xG
(G/xG). xG/S assesses the average quality of each shot a player attempted, and is referred to
as the quality of a shot hereafter. It is important to clarify that this performance measure has
nothing to do with the quality of the actual executions of the shots, but is rather a measure of
the average likelihood of scoring on the shots attempted. G/xG is a measure of how efficiently
a player converted his shots into goals.
Variables in the xG Model
Following the model building approach described in Section 4.2.2, the significant variables and
coefficients for the two types of shots are shown in Tables 5.1 and 5.2 for regular shots and free
kicks, respectively. The correlation coefficients between the variables can be seen in Table B.1
in Appendix B.
39
Chapter 5. Evaluating Results
Table 5.1: xG Model: Parameters for regular shots
Parameter
Constant
X1
X2
X3
X4
X5
X6
X7
XMS
XH/A
XElo
Description of Parameter
Distance to goal
Angle on goal
If shot occur after fast break
Number of take-ons before shot
If shot is executed by a header
If shot is assisted by cross
If shot is assisted by through ball
Current match status
Match location (home or away)
Difference in team strength
Coefficient Value
0.7417
0.1161
+0.7416
+0.4982
+0.4340
0.6883
0.5211
+0.9684
+1.3361
0.2700
0.0008
p-value
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
Parameter type
Continuous
Continuous
Binary
Discrete
Binary
Binary
Binary
Categorical
Binary
Continuous
Table 5.2: xG Model: Parameters for free kicks
Parameter Description of Parameter Coefficient Value
Constant
+9.1943
X1
Distance to goal
0.2730
X2
Angle on goal
11.8080
p-value
<0.01
<0.01
<0.01
Parameter type
Continous
Continous
These regression coefficients are inserted into Equation (3.4), estimating the likelihoods of
scoring on regular shots and free kicks, given a set of contextual variables. The two models,
in addition to the penalty conversion rate, which is equal to 0.7822, constitute the complete
xG Model for Tippeligaen. All shots attempted in 2014 and 2015 are evaluated and assigned a
likelihood of ending in a goal.
Consider the regression coefficients for regular shots in Table 5.1. In a football perspective
it makes sense that the distance, angle (measured in radians) and how a goal is executed are
parameters that are important for the conversion rate of shots, thereby supporting the values
on the coefficients of X1 , X2 , X3 , X5 , X6 and X7 . Distance and angle are highly negatively
correlated, and a short distance gives the keeper shorter time to react, while a large angle gives
a larger target to hit. A fast break is a situation where the defending team often do not manage
to reestablish balance, which might result in many attackers against fewer defenders than for
regular play. This is a reasonable explanation for the increase in the likelihood of scoring after
a fast break. A cross as assist or an execution by a header are both considered to be factors that
increase the difficulty of scoring a goal, yielding negative coefficients. Receiving a through ball
before taking the shot often results in fewer defenders between the player and the goal, which
increases the likelihood of scoring.
Some of the estimated coefficients from Table 5.1 are not easy to interpret at first look. It
might not be intuitive that the number of take ons (X4 ) before a shot increases the likelihood of
scoring. However, these are succeeded take ons, which might reduce the number of defenders
between the ball and the goal. The coefficients of current match status (XMS ), match location
(XH/A ) and difference in team strength (XElo ) also needs some explanation. The coefficient
of XMS is positive, hence, leading the match might increase the likelihood of converting a
shot. If a team is losing, they are likely to move the team further up the field, thereby possibly
leaving more space for the other team to attack. Furthermore, match location and difference
40
5.1 Expected Goals Model
in team strength have a similar impact on team tactics, but these variables have a negative
coefficient. For match location this might be because the home team on average has a more
offensive strategy, which can leave more space defensively. In addition, stronger teams have a
tendency to put pressure on the opponent, forcing them to a defensive stance. This can lead to
more defenders between the ball and the goal, decreasing the likelihood of converting a shot. A
comment is appropriate on the small value of the coefficient of team strength (XElo ) of 0.0008.
It is by far the smallest in magnitude, but the Elo difference for the teams in Tippeligaen ranges
from 350 to 350, and can therefore make a considerable impact on the shot values despite its
small coefficient.
As can be seen from Table 5.2, the significant variables in the model for free kicks are
distance to goal and angle on goal, which are both negative. Hence, an increase in distance
or angle decrease the likelihood of scoring. This is not surprising for distance to goal, but
may seem counter intuitive at first glance for the angle on goal. A possible explanation of the
high negative coefficient value can be seen in Figure 4.3(c), where it seems that a large share
of the free kicks attempted from the sides of the field, resulted in goals. However, a known
theory in football is that taking a free kick slightly off-centre is better than straight in front of
it. Having an angle could make it easier for the player to curve the ball when placing the free
kick, and it often forces the goalkeeper to choose a side to cover in the goal. The distance to
goal and the angle are negatively correlated, which means that an increase in distance reduces
the angle on goal, as seen in Figure 4.4. Because of the negative signs of the coefficients, this
leads to opposite impacts on the likelihood of scoring when moving further away from the goal.
However, it is believed that the negative coefficient of the distance can make the impact from
the angle on goal redundant when the angle becomes small enough. Therefore, the negative
coefficient of the distance can make the value of the angle on goal redundant, thus decreasing
the probability of scoring when moving the free kick away from the goal. After a thorough
consideration of these features of the model, the negative sign of the angle on goal coefficient
was accepted in the model.
When evaluating the two distinctive shot models, it is important comment on the data for
which the coefficient estimates are based on. Several of the parameters are subject to human
bias. Fast breaks, take ons and through balls are examples of parameters that are exposed to
subjective considerations, and it is a possibility that human bias might introduce error into the
coefficients. In addition, some of the parameters has been specified or calculated by the authors
based on the data set from Opta (see Table 4.3). This introduces another source of error in the
form of possible miscalculations.
5.1.1
Reliability
Table 5.3 shows the correlation coefficients of all five performance measures across the two
seasons. Only outfield players who played more than 900 minutes in both 2014 and 2015 are
considered, and the values are normalised per 90 minutes to evaluate players on equal terms.
S/90 min has the highest coefficient of 0.8811, and shows that players attempted a relatively
stable amount of shots per 90 minutes played across seasons. The relatively low correlation
coefficient of xG/S of 0.3570 indicates that the average quality of the shots a player attempted
across seasons was not necessarily the same.
Figure 5.1 shows the scatter plot of G/xG across the two seasons. This is the performance
measure used to rank players from the xG Model. It has the lowest correlation coefficient of all
five performance measures of 0.1067, which means that the consistency across the two seasons
41
Chapter 5. Evaluating Results
Table 5.3: xG Model: Correlation coefficients of performance measures from 2014 to 2015
G/90 min
0.6631
S/90 min
0.8811
xG/90 min
0.8494
xG/S
G/xG
0.3570 0.1067
was very limited. It indicates that players are not able to replicate their efficiency rates across
seasons. For comparison, players are much more consistent in both xG and in scoring goals,
which is reflected in the correlation coefficients of xG/90 min and G/90 min of 0.8494 and
0.6631, respectively.
Figure 5.1: xG Model: Scatter plot of the G/xG of all players with the correlation coefficient between
the 2014 and 2015 season
5.1.2
Validity
To assess the validity of the logistic regression models, consider the ROC curves for the two
shot models shown in Figure 5.2. For regular shots, the area under the ROC curve is equal to
0.84, as seen in Figure 5.2(a). This value is classified from the general rule introduced in Section
3.1 as excellent discrimination ability. Thus, the model for regular shots is good at assigning
observations with Y = 1 high likelihoods, and likewise low likelihoods to observations with
Y = 0. For the free kicks, the area under the curve is = 0.67, which is classified as right below
acceptable discrimination and can be seen in Figure 5.2(b).
42
5.1 Expected Goals Model
(a) Regular shots, area under ROC curve equal to 0.84
(b) Free kicks, area under ROC curve equal to 0.67
Figure 5.2: xG Model: ROC curve for the two shot models
This limited ability to discriminate between the outcomes of free kicks might suggest that
there is more to a free kick than only distance and angle. Only 558 free kicks are present in the
data set, and another season of data could possibly help to improve the discrimination abilities
of the model. Furthermore, a free kick is considered to be more dependent on the ability of the
player than for regular shots, for which no variables are tested. Despite these weaknesses, it is
believed that this model makes the likelihood estimation of the shots more accurate as opposed
to a simple free kick conversion rate. In addition, free kicks only constitute 4.3 % of the total
shots, and are therefore believed to have limited impact on the ratings for other players than
regular free kick takers.
5.1.3
Results
Table 5.4 shows all players scoring more than eight goals in Tippeligaen 2015, evaluated on
the five performance measures. The players in the table are ranked by G/xG. A value of G/xG
greater than one means that the player performed well, scoring more goals than expected according to the model. Conversely, if the value is below one, the player performed below expectations. For reference, 14 of the players in the table have been sold to foreign clubs during or
after the 2015 season.
The use of G/xG for rating players by efficiency is not unproblematic due to the sources of
error that exist. Most importantly, defender proximity is not accounted for, a variable deemed
significant by Lucey et al. (2014). Frequent goal scorers are possibly closer marked by defenders which could make their shots harder to convert. The free kick model introduces another
possible source of error to the xG values, due to its limited discrimination ability. Of the players in the table, Trond Olsen, Pål Alexander Kirkevold and Pål André Helland have the highest
share of shots coming from free kicks with 12 %, 9 % and 20 %, respectively. None of the
remaining players have a higher share than 5 %. Another important source of error is the randomness that exists in football. Goals are a rare events, and a single goal can affect the G/xG
value significantly. Extreme cases of players who scored few goals and seldom attempted shots
can also occur. An example is goalkeeper Håkon Opdal, who would have had a G/xG value of
around 2,400 because of his strike of luck on his only shot in 2015. However, this is to some
extent rectified by only including players scoring eight goals or more.
43
Chapter 5. Evaluating Results
Table 5.4: xG Model: All players scoring more than 8 goals in Tippeligaen 2015, ranked by G/xG
Name
Trond Olsen
Simon Diedhiou*
Veton Berisha*
Pål Alexander Kirkevold*
Kristoffer Ajer*
Marcus Pedersen
Alexander Sørloth*
Luc Kassi
Alexander Søderlund*
Tobias Mikkelsen*
Ernest Asante
Adama Diomandé*
Zdeněk Ondrášek*
Tommy Høiland
Matthı́as Vilhjálmsson
Mohamed Elyounoussi
Erling Knudtzon
Fredrik Nordkvelle
Iver Fossum*
Pål André Helland
Suleiman Abdullahi
Christian Gytkjær
Sander Svendsen
Fred Friday
Ola Kamara*
Bentley
Leke James*
Jón Dadi Bödvarsson*
Gustav Wikheim*
Olivier Occéan
*
Team
Position
Minutes played
Bodø/Glimt
Winger
2510
Haugesund
Forward
1945
Viking
Forward
1256
Sandefjord
Forward
1881
Start
Central midfielder
2581
Strømsgodset
Forward
835
Bodø/Glimt
Forward
1776
Stabæk
Forward
2130
Rosenborg
Forward
2242
Rosenborg
Winger
1931
Stabæk
Winger
2643
Stabæk
Forward
1850
Tromsø
Forward
2366
Molde
Forward
1103
Start/Rosenborg Forward
2087
Molde
Winger
2275
Lillestrøm
Forward
2595
Odd
Attacking midfielder 1891
Strømsgodset
Attacking midfielder 2653
Rosenborg
Winger
1502
Viking
Forward
1882
Haugesund
Forward
2545
Molde
Forward
1720
Lillestrøm
Forward
1770
Molde
Forward
2384
Odd
Winger
2464
Aalesund
Forward
2610
Viking
Forward
2067
Strømsgodset
Winger
2413
Odd
Forward
2208
sold to foreign club during or after the 2015 season
**
Goals
13
9
11
8
8
11
13
8
22
8
10
17
9
9
9
12
10
9
11
13
8
10
8
11
14
8
13
9
9
15
xG
6.55
4.90
6.57
5.35
5.43
8.20
9.89
6.26
17.44
6.44
8.06
13.92
7.57
7.93
7.99
10.69
8.91
8.06
10.07
12.07
7.77
9.84
8.13
11.22
14.69
8.53
13.97
10.06
11.35
20.23
Shots
74
52
47
66
43
34
58
54
87
69
70
79
74
36
46
84
51
48
69
89
70
57
60
67
89
67
91
58
68
89
xG/S
0.09
0.09
0.14
0.08
0.13
0.24
0.17
0.12
0.20
0.09
0.12
0.18
0.10
0.22
0.17
0.13
0.17
0.17
0.15
0.14
0.11
0.17
0.14
0.17
0.17
0.13
0.15
0.17
0.17
0.23
G/xG
1.98
1.84
1.67
1.50
1.47
1.34
1.31
1.28
1.26
1.24
1.24
1.22
1.19
1.13
1.13
1.12
1.12
1.12
1.09
1.08
1.03
1.02
0.98
0.98
0.95
0.94
0.93
0.89
0.79
0.74
on loan from a foreign club in 2015
Bodø/Glimt player Trond Olsen features on the top of the list for the 2015 season. He scored
13 goals, while the model expected him to score a mere 6.55 goals, which gives him a G/xG
value of 1.98. Nine of his 69 shots were from free kicks, which could introduce uncertainty
to his xG value. When examining only his regular shots, his G/xG value was 2.08. Simon
Diedhiou of Haugesund is ranked second in efficiency with a G/xG value of 1.84. Furthermore,
both Olsen and Diedhiou have an xG/S value of 0.09, which is the second lowest value of all.
For Olsen, this might be because he plays as a winger, thus might often attempt shots with low
angle on goal. The value of Diedhiou, on the other hand, is surprisingly low compared to his
striker colleague Christian Gytkjær. A closer look shows that the goals of Olsen had an average
xG of 0.15 and that he scored four goals with an xG below 0.04. The goals Diedhiou scored had
an average xG of 0.21, and two of his nine goals had xG below 0.10. This indicate that these
players converted low quality shots that influence their G/xG significantly. However, since the
correlation coefficient of G/xG is low across seasons, it is believed that Olsen and Diedhiou
might have a tough job replicating their efficiency in the future.
Other noticeable names on the list are Alexander Søderlund and Adama Diomandé, who
were number one and two on the top scorer list for 2015, respectively (see Table A.5). Søderlund
44
5.2 Markov Game Model 1
has slightly higher values in both xG/S and G/xG, as seen from the table. Further examination
showed that the goals of Søderlund had an average xG of 0.37, while Diomandés goals had an
average xG of 0.35. Thus, according to the model, the two forwards performed on a relatively
even level. However, these two players are considered good candidates of forwards that possibly
attract much attention from defenders, with both being Norwegian internationals. Because the
defender proximity variable is not accounted for in the model, their shots might be estimated as
easier than they actually were. Amongst the midfielders, the highest rated player is youngster
Kristoffer Ajer who played for Start, who finished 14th in the 2015 season (see Table A.4). He
is rated fifth in the model with a G/xG value of 1.47. When also taking into consideration his
position as a central midfielder, this efficiency rate is especially noticeable.
Among the players in the bottom of the table are familiar names like Olivier Occéan and
Gustav Wikheim. Occéan was the third highest goal scorer in 2015 with 15 goals, seemingly
performing on a high level for Odd. However, his efficiency rate was poor according to the
model, ranking him the least efficient player scoring eight goals or more, with G/xG equal
to 0.74. This might indicate that he attempted a high number of good quality shots, but did
not convert them efficiently. Further examination of Occéan showed that his goals on average
had an xG value of 0.51, which means that many of his goals were considered to come from
high quality shots. Former Strømsgodset winger Gustav Wikheim impressed many in the 2015
season, scoring 9 goals from his position as a winger. However, according to the model, his
efficiency in 2015 was relatively poor, with a G/xG of only 0.79. Furthermore, he had an xG/S
of 0.17, which is the highest among the wingers in Table 5.4. In addition, his goals had an
average xG of 0.40. However, these players should not be deemed inferior based on these
numbers. G/xG and xG/S shows low reliability across seasons, while xG/90 min has shown to
be more stable. Having high xG/90 min values compared to other players in the same positions,
it seems that these two players managed to create good scoring opportunities, and may yet be
valuable assets for their teams in the future.
The performance measure xG/S deserves a final comment. From Table 5.4, it seems that
the forwards on the teams that finished high on the table have the highest xG/S values. It might
be reasonable to suggest that teams with higher strength are more likely to get into favourable
positions for high quality shots, despite that the Elo variable accounting for team strength is
included in the model.
As mentioned in Section 4.2, a league table can be obtained by looking at the xG values
aggregated for each team. The league table obtained by looking at the difference in xG for all
teams is shown in Table B.5. Rosenborg won the league in 2015, and also tops the list in xG
difference, while Molde who finished sixth in the actual league table should have come second
had they performed as expected by the model.
5.2
Markov Game Model 1
Results from Model 1 are presented and discussed in this section. As described in Section
4.5, the reliability and validity of the model are discussed first, then the results from running
the model are presented and discussed. Top 10 lists for players in the following positions are
presented: forwards, wingers, attacking midfielders, central midfielders, full backs and centre
backs. The positions of all players are obtained from Transfermarkt (2016). Players are rated on
the basis of their average impact per 90 minutes played, and only outfield players who played
at least 900 minutes are considered. Goalkeepers are excluded because they are not suited for
45
Chapter 5. Evaluating Results
evaluation by this model. Player ratings from the 2014 season for the same positions can be
found in Appendix C.
5.2.1
Reliability
A scatter plot of the players who played over 900 minutes in both 2014 and 2015, a total of 116
players, and their respective values in the two years are shown in Figure 5.3.
Figure 5.3: Model 1: Scatter plot and correlation between the two seasons
As can be seen from Figure 5.3, the correlation coefficient between the values of each player
for the two seasons is positive. This points towards that the model might give high positive
values to the better players and not to players who occasionally did something extraordinary.
However, the correlation coefficient of 0.1944 might indicate that the reliability across the two
seasons is weak. Compared to the G/xG performance measure, the correlation for Model 1
across seasons is slightly higher. Further assessment of the correlation coefficient is difficult
because no similar models exists.
As can be seen from the scatter plot in Figure 5.3, some outliers are present in the data.
One of them is Veton Berisha, who is represented by the dot in the upper left corner. He went
from being rated as one of the worst players to one of the best in only one year. A closer look
revealed that the numerous shots he took in 2014 did not result in many goals (G/xG = 0.21),
and he was therefore punished by the model. In 2015, however, his efficiency was a lot better
(G/xG = 1.67) which is impacting his value positively.
5.2.2
Validity
Figures 5.4 and 5.5 show scatter plots of the value of each outfield player with their respective
rating from Altomfotball and VG. Altomfotball requires that a player has received ratings in at
least 20 matches to be included in the final list. Similarly, VG requires ratings from at least 18
46
5.2 Markov Game Model 1
matches. This lead to a comparison of 137 and 126 players with Altomfotball and 157 and 149
with VG in 2014 and 2015, respectively.
(a) 2014
(b) 2015
Figure 5.4: Model 1: Scatter plots and correlation with Altomfotball
(a) 2014
(b) 2015
Figure 5.5: Model 1: Scatter plots and correlation with VG-børsen
From Figures 5.4 and 5.5 it can be seen that the correlation coefficient is positive and larger
than 0.46 in all four cases. This indicates that the players that were considered the best by
the journalists (that is, scores a high average rating across a season), also were considered good
players by the model by delivering a high impact value per 90 minutes played. Some interesting
outliers are evident in the figures. In 2015, Mike Jensen was rated above 6 by both VG and TV2,
while scoring a value below 0.1 in the model, and is the upper left dot in both the figures for
2015. A closer examination of Mike Jensen is provided in Section 5.4.1. Another interesting
observation is the rightmost dot in 2015, which is Pål André Helland. Helland was rated as
the best player by the model and VG, but was only number 8 on the rankings provided by
Altomfotball.
47
Chapter 5. Evaluating Results
Figure 5.6 show scatter plots with the Transfermarkt values, which are given in the figure as
1,000 GBP. The players used for comparison consist of outfield players with over 900 minutes
on the field. This lead to a sample of 204 and 202 players in 2014 and 2015, respectively.
(a) 2014
(b) 2015
Figure 5.6: Model 1: Scatter plots and correlation with Transfermarkt
Again, from the figure it can be seen that the correlation coefficients are positive, but for
2014 it is as low as 0.1975. However, there is more to the market value of a player than his
performance over the last season. Factors such as age and nationality also influence the market
value to some extent (see for instance Sæbø (2015)). Moreover, the model has a bias towards
offensive players, but this is sometimes the case for market values as well (see Chapter 1).
Two outliers in the figure for 2015 are Papa Alioune Ndiaye and Ole Kristian Selnæs, who are
the two most valuable players at 3 and 2.25 million GBP, respectively. Veton Berisha, who is
only valued at 600 thousand GBP, is rated as number three by the model and represents the
bottom-right dot for 2015. For 2014, Berisha is the leftmost dot.
Table 5.5: Model 1: Correlations to VG-børsen, Altomfotball and Transfermarkt in 2014
VG-børsen
Altomfotball
Transfermarkt
Model 1
VG-børsen Altomfotball Transfermarkt Model 1
1
0.8111
1
0.4478
0.4345
1
0.5211
0.4602
0.1975
1
Tables 5.5 and 5.6 show the correlation matrix for the VG and Altomfotball ratings, Transfermarkt and Model 1. As can be seen from Table 5.6, the correlations between Altomfotball
and VG-børsen is as high as 0.8847 for the 2015 season. Model 1 has lower correlation with
Transfermarkt than with VG-børsen and Altomfotball in both seasons. This result is considered
to be expected, due to the numerous factors that most likely influence the market values from
Transfermarkt. In general, the correlations are higher for the 2015 season than for 2014.
However, Transfermarkt is considered the most objective benchmark of the three. Therefore
it is interesting to compare the models correlation with Transfermarkt, with the correlations
48
5.2 Markov Game Model 1
between Transfermarkt and the ratings by VG and Altomfotball. For both 2014 and 2015 the
ratings provided by VG and Altomfotball show a higher correlation with Transfermarkt than the
model. A portion of this is believed to be due to a larger bias towards offensive players in the
model compared with VG and Altomfotball.
Table 5.6: Model 1: Correlations to VG-børsen, Altomfotball and Transfermarkt in 2015
VG-børsen
Altomfotball
Transfermarkt
Model 1
VG-børsen Altomfotball Transfermarkt Model 1
1
0.8847
1
0.6693
0.6684
1
0.4987
0.5182
0.3499
1
As a closing comment, the fact that Model 1 shows positive correlations with all three
benchmarks is considered as favourable for the validity, despite none of them being a fully
comparable source. All three are believed to capture player performance to some extent, which
is the aim of Model 1. Players that have been sold to foreign clubs after the 2015 season are
highlighted for reference, not to serve as validation but rather as observations of what players
attracted attention from clubs outside Norway.
5.2.3
Results
The way Model 1 assigns values to shots is important to keep in mind for the results presented
below. As described in Section 4.3, two types of artificial shot events are created in order to
keep track of the outcomes of the shots. This means that for each state, the state-action pair
corresponding to a shot in this state, Q(s, Shot), is assigned a value equal to the likelihood
of scoring when being in this state. With the impact function in Equation (4.5), a player can
receive two different values when shooting from a given state. If he scores he would receive
I = 1 Q(s, Shot), and if he misses he would receive I = (⇡ 0) Q(s, Shot). The first
number in the latter expression is not equal to zero, since there is some value to missing a shot
due to the possibility of a rebound. For instance, if the likelihood of scoring in a given state is
0.15, a player would receive an impact of I = 1 0.15 = 0.85 if he scores and I ⇡ 0.15 if
he misses. This makes shots and goals heavy contributors to the value of players in this model.
However, if a player takes a lot of shots and rarely scores he will often get punished with
negative values and thereby a lower rating. Hence, the values of the shots resemble G/xG. This
is not an unreasonable way of being evaluated for forwards, but for players in other positions,
where scoring goals is not the primary interest, it might be unfair.
Top 10 lists for the different positions are presented in Tables 5.7 to 5.12. Similar for all the
tables are player names, minutes played and total impact value per 90 minutes. In addition, they
show the total impact from shots and the five highest valued actions for the respective positions
on average. When referring to the average value in the discussion below, it is always regarding
the average value of the top 10 players for the position in question.
Consider the top 10 forwards shown in Table 5.7. The primary task of a forward is to score
goals, and they are normally not as involved in the build up play as players further back on the
field. Passes and other non-shot involvements are therefore less numerous, and the number of
shots attempted is higher.
49
Chapter 5. Evaluating Results
Table 5.7: Model 1: Top 10 forwards, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Adama Diomandé*
Veton Berisha*
Alexander Søderlund*
Tommy Høiland
Fred Friday
Alexander Sørloth*
Luc Kassi
Erling Knudtzon
Simon Diédhiou*
Matthı́as Vilhjálmsson
Top 10 average
*
Team
Minutes
Stabæk
1850
Viking
1256
Rosenborg
2242
Molde
1103
Lillestrøm
1770
Bodø/Glimt
1776
Stabæk
2130
Lilestrøm
2595
Haugesund
1945
Start/Rosenborg 2087
sold to foreign club during or after the 2015 season
**
Total
0.5448
0.4721
0.3896
0.3771
0.3130
0.3108
0.2759
0.2481
0.2327
0.1961
0.3360
Shot
0.2736
0.3702
0.3674
0.2945
0.1145
0.2304
0.1064
0.1051
0.1287
0.0941
0.2085
Pass
0.0252
0.0475
-0.0244
0.0077
-0.0011
-0.0481
0.1279
0.0665
0.0097
0.0450
0.0256
Take on
0.0287
-0.0167
0.0003
0.0175
0.0574
0.0258
0.0156
0.0139
0.0285
0.0047
0.0175
Foul won
0.0317
0.0063
0.0094
0.0045
0.0207
0.0154
0.0140
0.0118
0.0129
0.0060
0.0133
Cross
0.0454
0.0097
0.0047
0.0090
0.0034
0.0091
0.0051
0.0030
-0.0224
0.0146
0.0082
Carry
0.0519
0.0562
0.0154
0.0099
0.0954
0.0250
-0.0036
0.0374
0.0431
0.0317
0.0362
on loan from a foreign club in 2015
According to Model 1, Adama Diomandé was the best forward in Tippeligaen 2015. He
obtained most of his total value from accurate shooting, which resulted in 17 goals in 2015,
in addition to scoring above average in five out of six categories. Similar observations can be
made regarding the shot value for most of the other forwards, especially top scorer Alexander
Søderlund, who seems to have limited impact on games beside scoring goals. This illustrates
that these two forwards, who were the top two goalscorers, seem to be quite different players.
An exception to the observation regarding the impact of shots, is Fred Friday. As seen, his
value for shots was approximately the same as for his ball carries. He also had the highest
impact from take ons for the players in the table. Another exception is Luc Kassi, who had
a higher impact from passes than for shots. These observations show the importance of shots
in Model 1, but it seems that forwards with good passing abilities and dribbling skills also
are rewarded. One noticeable player that is missing in the table is Marcus Pedersen, who was
bought by Strømsgodset in August 2015. Pedersen scored 11 goals on his 10 games that fall,
and would have been at the top of the list of forwards if he had played enough minutes (he only
played 835 minutes).
Table 5.8: Model 1: Top 10 wingers, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Team
Pål André Helland
Rosenborg
Trond Olsen
Bodø/Glimt
Moryké Fofana*
Lillestrøm
Zymer Bytyqi
Viking
Ernest Asante
Stabæk
Gustav Wikheim*
Strømsgodset
Espen Børufsen
Start
Ole Jørgen Halvorsen
Odd
Magnus Andersen
Tromsø
Mohamed Elyounoussi Molde
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Minutes
1502
2510
1300
1258
2643
2413
2154
1393
2663
2275
Total
0.5553
0.3749
0.3683
0.3111
0.2961
0.2935
0.2209
0.2160
0.1950
0.1855
0.3017
Shot
0.2431
0.1932
0.1777
0.0527
0.0470
-0.0084
0.0815
0.0144
0.0428
0.0892
0.0933
Pass
0.0381
0.0374
0.0728
0.0236
0.0850
0.1016
-0.0075
0.0047
0.0321
0.0513
0.0439
Take on
0.0055
0.0244
0.0400
0.0130
0.0226
0.0708
0.0107
-0.0057
-0.0030
-0.0051
0.0173
Foul won
0.0357
0.0110
0.0109
0.0043
0.0172
0.0037
0.0084
0.0038
0.0036
0.0185
0.0117
Cross
0.0021
0.0503
-0.0013
0.1469
0.0206
0.0528
0.0619
0.1064
0.0343
-0.0146
0.0459
Carry
0.1153
0.0503
0.0098
0.0315
0.0741
0.0439
0.0145
0.0295
0.0374
0.0165
0.0423
on loan from a foreign club in 2015
Table 5.8 shows the top 10 wingers. Wingers play on the sides of the field and are normally
50
5.2 Markov Game Model 1
heavily involved in crossing and should be good in one-on-one situations to be able to get into
good crossing positions. The impact function used to value the player involvements, Equation
(4.5), takes into account what a given action lead to. For the wingers, this has at least one
important consequence to be aware of. A cross that resulted in a shot from a teammate is
valued higher than a cross that resulted in for example an interception by the opposing team.
This means that a cross can be perfectly hit into the area from the winger without him getting
full reward if the finisher did not do a good job. As pointed out in Secion 4.3.3, this way of
rewarding is believed to be fair in the long run.
Pål André Helland tops the list of wingers, and he is in fact the highest rated player of all
in Model 1. Helland also tops the player ratings from VG (2015). His total impact value comes
primarily from shots and ball carries, where he has the two highest values in the table. On the
other hand, it is surprising to see his low impact value from crosses. Trond Olsen, considered
by the xG Model as the most efficient player, is second on the list. Besides from shots, most of
his impact came from crosses and ball carries which are important for players in his position.
The two players with the highest impact from crosses were Ole Jørgen Halvorsen and Zymer
Bytyqi. They also had below average impact from shots, and their crossing abilities seemed
to be their primary contribution. The only winger with negative impact from shots is Gustav
Wikheim, but his total value came primarily from passes and take ons where he has the highest
impact values in the table. His high impact value from passes might imply that he chooses to
pass more often than the others.
As for the forwards, shots are still a heavy contributor to the total impact value of the
wingers. However, it seems that Model 1 is able to give value to wingers with good crossing abilities and to players that are good at passing and take ons.
Now consider the top 10 attacking midfielders shown in Table 5.9. Attacking midfielders
are expected to deliver key passes and assists to the forwards, in addition to being a threat to
the opposition goal from long range shooting. They can have fewer defensive tasks than central
midfielders, and are often technically gifted players with good passing and dribbling abilities.
Table 5.9: Model 1: Top 10 attacking midfielders, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Team
Fredrik Nordkvelle
Odd
Daniel Fredheim Holm
Vålerenga
Michael Barrantes*
Aalesund
Eirik Hestad
Molde
Papa Alioune Ndiaye*
Bodø/Glimt
Ghayas Zahid
Vålerenga
Iver Fossum*
Strømsgodset
Gjermund Åsen
Tromsø
Henrik Furebotn
Bodø/Glimt
Thomas Kind Bendiksen* Molde
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Minutes
1891
1974
1000
941
1286
2383
2653
1709
2157
931
Total
0.2719
0.2687
0.2261
0.2161
0.2087
0.2055
0.2045
0.2026
0.1731
0.1407
0.2118
Shot
0.1954
0.1749
0.0503
0.0232
-0.0279
0.0082
0.0744
-0.0156
0.0769
-0.0503
0.0509
Pass
0.0041
0.0404
0.0358
0.0963
0.0383
0.0851
0.0371
0.0095
-0.0008
0.0233
0.0369
Cross
0.0174
0.0097
-0.0020
0.0041
0.0189
-0.0040
0.0174
0.0334
0.0395
0.0156
0.0150
Corner
-0.0011
0.0026
0.0122
0.0195
0.0098
-0.0007
-0.0003
0.0814
0.0030
0.0900
0.0216
Carry
-0.0069
-0.0116
0.0397
-0.0072
0.0321
0.0479
0.0226
0.0355
0.0047
0.0084
0.0165
FK
-0.0006
0.0021
0.0297
0.0381
-0.0010
-0.0030
0.0277
0.0241
0.0332
0.0150
on loan from a foreign club in 2015
For the two players on top, Fredrik Nordkvelle and Daniel Fredheim Holm, the total impact
value came primarily from shots. They scored nine and seven goals respectively, and had by
far the highest impact value from shots of all the attacking midfielders. Third placed Michael
Barrantes had a much lower impact value from shots, but obtained a significant portion of
51
Chapter 5. Evaluating Results
his impact value from ball carries and by executing free kicks accurately. Eirik Hestad had
the highest impact value from passes, in close race with Ghayas Zahid, where the latter also
seems to have performed well on ball carries. Fifth place on the list, Papa Alioune Ndiaye, is
a player with no particularly large impact values. However, he was above average in many of
the categories not shown in the table like long passes, take ons and ball recoveries. This is a
nice example of how Model 1 is able to give a high total value to players performing well on a
variety of actions, like seen earlier for Fred Friday and Gustav Wikheim.
Next, consider the top 10 central midfielders in Table 5.10. Playing as a central midfielder
requires a wide set of skills, including passing, duel play, tackling and strong physical condition.
They are less likely to get into scoring positions than the attacking midfielders, which might
influence the impact values of these players. A more defensive role also means that they more
seldom play passes that result in shots or other goal scoring opportunities. Nonetheless, the
central midfielders who regularly wins tackles, intercepts, wins duels and is accurate in his
passing, should be rewarded in the model.
Table 5.10: Model 1: Top 10 central midfielders, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Team
Minutes
Christian Grindheim
Vålerenga
2675
Malaury Martin
Lillestrøm
975
Giorgi Gorozia
Stabæk
1467
Kristoffer Ajer*
Start
2581
Bismark Adjei-Boateng** Strømsgodset 1293
Morten Konradsen
Bodø/Glimt
1362
Kamal Issah
Stabæk
1846
Fredrik Midtsjø
Rosenborg
2396
Johan Andersson
Lillestrøm
973
Ole Kristian Selnæs*
Rosenborg
1951
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Total
0.3087
0.2947
0.2326
0.2187
0.1847
0.1740
0.1733
0.1718
0.1664
0.1553
0.2080
Shot
0.0985
0.1446
-0.0578
0.1016
0.0590
0.1439
0.0274
0.0227
0.1054
0.0157
0.0661
Pass
0.1239
0.0453
0.1369
0.0243
0.0192
-0.0058
0.0838
0.0833
0.0429
0.0782
0.0632
Tackle
0.0069
0.0045
0.0065
0.0077
0.0214
0.0044
0.0143
0.0152
0.0046
0.0133
0.0099
Ball rec
0.0105
0.0088
0.0241
0.0122
0.0026
0.0071
0.0163
0.0118
0.0117
0.0217
0.0127
Cross
0.0040
0.0103
0.0063
0.0066
0.0440
0.0202
0.0082
0.0010
-0.0056
0.0226
0.0118
Corner
0.0125
0.0336
0.0858
-0.0002
-0.0009
-0.0014
-0.0025
0.0127
on loan from a foreign club in 2015
Christian Grindheim of Vålerenga is the highest rated central midfielder. He had an above
average shot value, and his impact from passes is the second highest in the table. This is not
surprising considering his five goals and eleven assists in 2015. Third place on the list, Giorgi
Gorozia actually had a negative impact value from shots. However, he had the highest impact
value for passes, ball recoveries and corners of the players in the table. Bismark Adjei-Boateng
had the highest impact from tackles and also was the best to hit crosses amongst the central
midfielders. As was the case for the attacking midfielders, players with no exceptionally high
values are rated high, like Rosenborg players Fredrik Midtsjø and Ole Kristian Selnæs, are also
appreciated by Model 1. The latter had the second highest impact value from ball recoveries,
for which he was praised by pundits and journalists throughout the season. Good examples of
players that are on the list due to their impact from shots, are Morten Konradsen and Johan
Andersson.
A noticeable player that is missing on the list of central midfielders is RBK player Mike
Jensen. He was considered by many pundits and journalists as the best player in 2015. Altomfotball (2015) regarded him as the best player, while according to VG (2015) he was second
only to Pål André Helland. A closer investigation on why he is not present on the list revealed
that he took a lot of shots and scored few goals. He attempted 81 shots, seventh overall in Tip52
5.2 Markov Game Model 1
peligaen and scored only three goals, which lead to a G/xG equal to 0.51. As described earlier,
this is very unfavourable with this model specification and makes him drop well out of top 10.
Not even his 13 assist are enough to get him on the list. To illustrate how significant an impact
his shots had on his ranking; if shots were excluded from the model, he would have been the
second best central midfielder. A case study of the involvements Mike Jensen had in 2014 and
2015 is shown in Section 5.4.1.
Table 5.11 shows the top 10 full backs in Tippeligaen 2015. Model 1 has some flaws when
it comes to valuing players in defensive positions. Due to how the data is built with events
around the ball, it is not possible to capture when a full back or centre back should have been
in position to interfere but was not. This makes it harder to punish defenders for mistakes. In
addition, they often get punished for clearing the ball to corner and throw-ins, typically where
state transitions to the other team occur. However, when players in the same positions are
compared, it is possible to say something about the relative performance of players.
A modern full back has both defensive and offensive responsibilities. In defence, they
should be good in one-on-one situations, tackling and positional play setting up possible interceptions. Solid passing ability is also an important aspect for the full backs, especially if his
team plays a possession-oriented style. In attacking play, the full backs are often combining
with the wingers and attacking midfielders to get into good crossing positions.
Table 5.11: Model 1: Top 10 full backs, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Per-Egil Flo
Espen Ruud
Lars-Christopher Vilsvik
Jo Nymo Matland*
André Danielsen
Kent-Are Antonsen
Joachim Olsen Solberg
Zarek Chase Valentin*
Jørgen Skjelvik
Mikael Dorsin
Top 10 average
*
Team
Molde
Odd
Strømsgodset
Aalesund
Viking
Tromsø
Mjøndalen
Bodø/Glimt
Rosenborg
Rosenborg
sold to foreign club during or after the 2015 season
**
Minutes
1998
2476
2126
1425
2700
2248
2486
2074
1812
1582
Total
0.3290
0.2953
0.2692
0.2429
0.2329
0.2275
0.2087
0.2006
0.1855
0.1492
0.2341
Shot
0.0396
0.0457
0.0542
0.0605
0.0995
0.0557
-0.0588
0.0056
0.0154
0.0365
0.0354
Pass
0.1003
0.0925
0.0197
-0.0015
0.0428
0.0029
0.0230
0.0112
0.0801
0.0294
0.0400
Cross
0.0327
0.0831
0.1462
0.0426
0.0459
0.0277
0.0192
0.0498
0.0449
0.0210
0.0513
Ball rec
0.0092
0.0070
0.0134
0.0049
0.0155
0.0160
0.0143
0.0114
0.0120
0.0106
0.0114
Corner
0.0700
0.0087
0.0153
0.0580
0.0179
0.0102
0.0508
-0.0025
0.0070
-0.0011
0.0234
FK
0.0087
0.0204
-0.0021
0.0501
0.0050
0.0261
0.1031
0.0014
0.0006
-0.0011
0.0212
on loan from a foreign club in 2015
Per-Egil Flo is the highest rated full back in Tippeligaen 2015. As can be seen in Table D.2,
Flo came out second in 2014, which shows he has been performing consistently on a high level.
He had the highest impact value from passes of the players in the table, and his corner kicks
also contributed significantly to his total value. Interestingly, his value from crossing is below
average. Number two on the list, Espen Ruud also had a high impact from passing, but unlike
Flo had an above average value from crosses amongst the full backs. Lars-Christopher Vilsvik
is the full back with the best impact value from crosses by a wide margin.
All of the top three full backs are considered good offensive players, and this is supported by
the results from Model 1. As one might expect, none of the full backs, except André Danielsen,
got a high impact from shots. Additionally, a noticeable player missing is Rosenborg player
Jonas Svensson. He was rated 8th and 6th in VG (2015) and Altomfotball (2015) respectively,
but he is not in the list for the same reason as for Mike Jensen. He did not score any goals in
the 2015 season, while attempting 25 shots.
53
Chapter 5. Evaluating Results
Finally, consider the centre backs in Table 5.12, which are the core of the defence in football
teams. They should be good at tackling, aerial duels and positional play in order to stop the
attacks coming from the opposition. Again, the way the impact function is specified, actions
such as interceptions, clearance and aerial duels are rewarded on the basis of what they lead to.
Table 5.12: Model 1: Top 10 centre backs, Tippeligaen 2015.
Average total value = 0.1124, Minimum total value = -0.1248
Player
Johan Bjørdal
Rhett Bernstein*
Joona Toivio
Andreas Nordvik*
Brede Moe
Lars-Kristian Eriksen
Ole Christoffer Heieren Hansen
Vegard Forren
Morten Sundli
Steffen Hagen
Top 10 average
*
Team
Rosenborg
Mjøndalen
Molde
Sarpsborg 08
Bodø/Glimt
Odd
Sarpsborg 08
Molde
Mjøndalen
Odd
sold to foreign club during or after the 2015 season
**
Minutes
1093
1234
1706
1846
2408
2340
2059
2416
2018
2610
Total
0.2398
0.1927
0.1781
0.1667
0.1657
0.1456
0.1383
0.1196
0.1050
0.1000
0.1551
Shot
0.0461
0.1214
-0.0190
0.0532
0.0544
0.0235
0.0634
-0.0375
0.0314
0.0174
0.0354
Pass
0.1655
0.0301
0.0665
0.0586
0.0545
0.0746
0.0578
0.0902
0.0381
0.0813
0.0717
Clearance
-0.0001
-0.0068
0.0274
0.0168
0.0304
0.0132
0.0337
0.0199
-0.0137
-0.0033
0.0117
Ball rec
0.0107
0.0077
0.0093
0.0065
0.0072
0.0084
0.0051
0.0069
0.0092
0.0089
0.0080
Cross
-0.0026
0.0002
0.0650
0.0036
-0.0016
0.0003
0.0000
0.0052
0.0029
-0.0002
0.0073
Long pass
0.0012
-0.0038
0.0125
0.0403
-0.0082
0.0110
-0.0025
0.0256
-0.0034
0.0079
0.0081
on loan from a foreign club in 2015
RBK defender Johan Bjørdal was the best central defender according to Model 1. Surprisingly, he had the highest impact value from passes of all players in Tippeligaen 2015, and is well
clear of his centre back colleagues in this area. His value from ball recoveries also contributed
to his total impact value. Rhett Bernstein was the highest goal scorer amongst the centre backs,
which is reflected in his impact value for shots. Andreas Nordvik had the highest impact value
from long passes, which might be an important attribute of a centre back, while Ole Christoffer
Heieren Hansen had the highest value for clearances. For the defenders, it seems that Model
1 is to some extent able to appreciate defensive attributes. Furthermore, the impact values are
smaller for the defensive actions, hence, making shots, passes and other more offensive actions
heavier contributors.
As mentioned in Section 4.3.4, it is possible to form a league table based on the results
from the Markov game models. Results from doing this for Model 1 are shown in Appendix C,
Tables C.3 and C.6 for 2014 and 2015, respectively. In both the tables, the estimated table and
the real table does not differ much. This is not very surprising due to the heavy effect shots have
on the values. However, Table C.6 can once again indicate that Molde underachieved compared
to what could be expected in 2015.
54
5.3 Markov Game Model 2
5.3
Markov Game Model 2
The results from Markov Game Model 2 are presented and discussed in this section. The structure of the section is the same as for Model 1 in the previous section. Again, only outfield
players who played at least 900 minutes are considered. Player ratings from the 2014 season
can be found in Appendix D.
5.3.1
Reliability
A scatter plot of the players who played over 900 minutes in both 2014 and 2015, a total of 116
players, and their respective values in the two years are shown in Figure 5.7.
Figure 5.7: Model 2: Scatter plot and correlation between the two seasons
From Figure 5.7 it can be seen that the correlation coefficient is positive, like for Model
1. However, with a value of 0.8988, the coefficient is substantially higher for Model 2. The
coefficient shows excellent correlation across the two seasons, and thus the model seems to be
very stable. As mentioned earlier, this might indicate that the model is favoring the best players
instead of assigning high values to players that occasionally have a big impact on the matches.
Like for Model 1, no similar models are available for comparison. The dot in the upper right
corner on the figure represents Pål André Helland, who performed very well in both seasons
according to the model. The rightmost dot is Trond Olsen, who in this model seemed to have a
very good season in 2014, while 2015 was more average. It is worth mentioning that the results
for Trond Olsen in Model 1 is opposite, his 2014 season was average while he was very good
in 2015.
55
Chapter 5. Evaluating Results
5.3.2
Validity
Figures 5.8 and 5.9 shows scatter plots of the value of each player with their respective rating
from Altomfotball and VG. The players considered in these figures, are the same as for Model
1.
(a) 2014
(b) 2015
Figure 5.8: Model 2: Scatter plots and correlation with Altomfotball
(a) 2014
(b) 2015
Figure 5.9: Model 2: Scatter plots and correlation with VG-børsen
From Figures 5.8 and 5.9 it is evident that the correlation is positive in all four cases. The
correlation coefficients are a bit lower in two out of four cases compared to Model 1. All the
coefficients have values above 0.42 nonetheless, which yields the same conclusion as for the
previous model. Players that are considered best by the journalists also tend to be considered
good by the model. Some outliers in the figures are worth mentioning. The rightmost dot
in both Figure 5.8(a) and Figure 5.9(a) is Trond Olsen, who performed very well according
to the model while only achieving average ratings by the media. As previously pointed out,
Altomfotball rated Pål André Helland as number 8 in 2015, while VG had him as number one.
This can be seen in the figures for 2015, where Helland is the rightmost dot. One last player that
56
5.3 Markov Game Model 2
is worth pointing out is Martin Ødegaard who represents the upper right dot in Figure 5.9(a).
Ødegaard did not play enough to be included in the final list by Altomfotball, and is therefore
not included in Figure 5.8(a).
The scatter plots and comparison with Transfermarkt values can be seen in Figure 5.10.
(a) 2014
(b) 2015
Figure 5.10: Model 2: Scatter plots and correlation with Transfermarkt
It can be seen from Figure 5.10 that the correlation coefficients are positive for both seasons.
Model 2 shows improvements compared to Model 1, yiedling higher correlation coefficients for
both seasons. As pointed out earlier, there is more to the market value of a player than their
performance in a given season. Again, Papa Alioune Ndiaye and Ole Kristian Selnæs are two
conspicuous outliers with their high market values in 2015. The two highest market values
in 2014, were defenders Lars-Christopher Vilsvik and Vegard Forren. It seems that Model 2
assigned average impact values to the two, and this is possibly due to the models bias towards
offensive involvements and players, which is discussed further in the next section.
Table 5.13: Model 2: Correlations between VG-børsen, Altomfotball, Transfermarkt and Model 1 in
2014
VG-børsen
Altomfotball
Transfermarkt
Model 1
Model 2
VG-børsen
1
0.8111
0.4478
0.5211
0.4218
Altomfotball
Transfermarkt
Model 1
Model 2
1
0.4345
0.4602
0.4354
1
0.1975
0.2506
1
0.3597
1
As for Model 1, the correlation matrix for Model 2 to the three benchmarks is shown in
Table 5.13 and 5.14. Again in general, the correlations in 2014 are lower than in 2015. This is
also the case for the correlations between the two models. The biggest difference from Model
1 is the correlations with the Transfermarkt values. As was the case for Model 1, VG and
Altomfotball achieves a higher correlation with Transfermarkt than Model 2. Again it is worth
mentioning that this is believed to be mainly due to a larger bias towards offensive players in the
57
Chapter 5. Evaluating Results
models than in the ratings by VG and Altomfotball. The positive correlations between Model
2 and the three benchmarks, is considered as favourable for the validity. All three benchmarks
are believed to capture player performance to some extent, which is also the aim of Model 2.
Table 5.14: Model 2: Correlations between VG-børsen, Altomfotball, Transfermarkt and Model 1 in
2015
VG-børsen
Altomfotball
Transfermarkt
Model 1
Model 2
VG-børsen
1
0.8847
0.6693
0.4987
0.5785
Altomfotball
Transfermarkt
Model 1
Model 2
1
0.6684
0.5182
0.5221
1
0.3499
0.4685
1
0.5567
1
Like for Model 1, players that have been sold to foreign clubs after the 2015 season are
highlighted for reference only.
5.3.3
Results
The way Model 2 assigns values to players is important to keep in mind when interpreting the
results presented below. Because of how a state is defined for Model 2, described in Section 4.4,
the value of a state is actually the value of that specific involvement. This has some implications
for how players are given impact, and especially for shots. By the definition of the impact
function, I(s) = Q(s), a player is assigned the value of attempting a shot, regardless of whether
it ended in a goal or not. Hence, the values for shots in this model resembles that of xG to a
larger extent than G/xG. This results in almost entirely positive shot values, and hence, the
punishment for missing shots, which was evident in Model 1, is limited.
Like for Model 1, top 10 lists for the different positions are presented in Tables 5.15 to
5.20. Similar for all the tables are player names, minutes played and total impact value per 90
minutes. In addition, they show the total impact from shots and the five highest valued actions
for the respective positions on average. When referring to the average value in the discussion
below, it is always regarding the average value of the top 10 players for the position in question.
Table 5.15 shows the top 10 lists for forwards in Tippeligaen 2015. Adama Diomandé is on
top of the list, just as he was in Model 1. As can be seen from the table, he obtained most of
his total impact value from shots, passes and ball carries. His value for take ons is also above
average, which can indicate that Diomandé is a forward who performed well on a number of
important attributes in his position. Olivier Occéan is number two on the list. He had a higher
impact value from his shots than Diomandé (reflected in his high xG value in Table 5.4) and his
impact value from aerial duels is the highest for the forwards in the table. Third placed Fred
Friday had the highest impact value from both take ons and ball carries, which indicate he is
good at bringing the ball into favourable positions and has the ability to dribble past defenders.
Top scorer Alexander Søderlund is ninth on the list. He had a high impact value from shots, but
is below average in all other categories except aerial duels. This further supports the suspicion
that he did not contribute much in the build up play, but focused on scoring goals. In total, five
out of top 10 are the same as for Model 1.
The top 10 wingers are shown in Table 5.16, where Pål André Helland is rated on top. In
fact he is considered the best player of all in Model 2. As can be seen from the table, he had
58
5.3 Markov Game Model 2
Table 5.15: Model 2: Top 10 forwards, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Team
Minutes
Adama Diomandé*
Stabæk
1850
Olivier Occéan
Odd
2208
Fred Friday
Lillestrøm
1770
Ola Kamara*
Molde
2384
Veton Berisha*
Viking
1256
Sander Svendsen
Molde
1720
Jón Dadi Bödvarsson* Viking
2067
Leke James*
Aalesund
2610
Alexander Søderlund* Rosenborg 2242
Alexander Sørloth*
Bodø/Glimt 1776
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Total
1.0543
0.9867
0.9133
0.9085
0.8831
0.8309
0.8080
0.8047
0.7993
0.7916
0.8781
Shot
0.5001
0.5049
0.4026
0.4298
0.3770
0.3897
0.3123
0.3676
0.4564
0.3829
0.4123
Pass
0.2130
0.2432
0.1939
0.2208
0.2317
0.1975
0.1932
0.1920
0.1677
0.1769
0.2030
Take on
0.0656
0.0128
0.0740
0.0143
0.0165
0.0471
0.0326
0.0487
0.0131
0.0233
0.0348
Foul won
0.0333
0.0407
0.0303
0.0246
0.0232
0.0101
0.0273
0.0212
0.0279
0.0265
0.0265
Aerial
0.0312
0.0888
0.0115
0.0210
0.0258
0.0047
0.0365
0.0609
0.0491
0.0719
0.0401
Ball carry
0.1160
0.0616
0.1419
0.1069
0.1247
0.1174
0.1328
0.0723
0.0550
0.0762
0.1005
on loan from a foreign club in 2015
the highest and third highest impact values from shots and passes, respectively. He also had the
highest value from ball carries in the list. This might show why he was considered a vital part
in the Rosenborg team winning a domestic double in 2015. Second on the list, Gustav Wikheim
had a below average impact value from shots, and the highest impact value from passes and
take ons. In addition, he had an above average value for ball carries, which was also observed
in the results for Model 1. Ninth on the list is Odd player Bentley, who had the highest impact
value from crosses of the players in the list, just ahead of his teammate Ole Jørgen Halvorsen.
This corresponds well with the fact that their teammate Olivier Occéan had the highest impact
value from aerial duels among the forwards. These two are below average on all other attributes,
which is reflected in their position on the list.
Table 5.16: Model 2: Top 10 wingers, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Team
Pål André Helland
Rosenborg
Gustav Wikheim*
Strømsgodset
Yassine El Ghanassy* Stabæk
Mohamed Elyounoussi Molde
Tobias Mikkelsen*
Rosenborg
Moryké Fofana*
Lillestrøm
Ernest Asante
Stabæk
Samuel Adegbenro
Viking
Bentley
Odd
Ole Jørgen Halvorsen
Odd
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Minutes
1502
2413
1879
2275
1931
1300
2643
1778
2464
1393
Total
1.4047
1.1020
0.9840
0.9445
0.9015
0.8512
0.8326
0.8245
0.8027
0.7688
0.9416
Shot
0.4942
0.3125
0.2875
0.3526
0.3444
0.2796
0.2662
0.3356
0.2561
0.2770
0.3206
Pass
0.2508
0.3692
0.2355
0.2887
0.2291
0.2384
0.2453
0.1451
0.1901
0.1771
0.2369
Take on
0.1021
0.1477
0.1028
0.0509
0.0446
0.1325
0.0436
0.0648
0.0436
0.0272
0.0760
Corner won
0.0379
0.0246
0.0242
0.0213
0.0197
0.0241
0.0195
0.0212
0.0294
0.0271
0.0249
Cross
0.0729
0.0381
0.0531
0.0255
0.0630
0.0176
0.0683
0.0554
0.1292
0.1195
0.0643
Ball carry
0.1821
0.1562
0.1261
0.1259
0.1587
0.0852
0.1435
0.1234
0.0829
0.0823
0.1266
on loan from a foreign club in 2015
Examples of players that did not obtain any particularly high impact values, are third and
fifth placed Yassine El Ghanassy and Tobias Mikkelsen. This can show that Model 2 is able to
value players with a wide spread of abilities to a certain extent, also to players rated high on
the lists. It can be seen that these two players have impact values around the average in all six
categories. Six of the wingers in Table 5.16 are the same as for Model 1. A noticeable player
missing is Trond Olsen, who was considered the most efficient player by the xG Model and
59
Chapter 5. Evaluating Results
rated second among the wingers in Model 1. Although he was efficient, his xG value was not
particularly high, which makes him drop out of the list for Model 2.
The top 10 list for attacking midfielders can be found in Table 5.17. Iver Fossum is rated on
top. He had the highest and second highest impact value of the players in the list in passes and
ball carries, respectively. In addition, his value from shots is also above average. Papa Alioune
Ndiaye has the second highest shot value, and the best value for take ons. As previously seen,
he had the highest market value from Transfermarkt at year end 2015, at 3 million GBP. Tenth
on the list, youngster Eirik Hestad is an example of a player who obtained most of his impact
value from other actions than shots. He has by far the lowest value in this action amongst the
players in the list, and he earned most of his value by passing, corner kicks and ball carries.
He is possibly a player to watch in the future, considering the fact that he is only 20 years old.
Eight of the attacking midfielders in Table 5.17 also appeared on the top 10 list from Model 1.
Table 5.17: Model 2: Top 10 attacking midfielders, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Team
Iver Fossum*
Strømsgodset
Papa Alioune Ndiaye* Bodø/Glimt
Ghayas Zahid
Vålerenga
Michael Barrantes*
Aalesund
Fredrik Nordkvelle
Odd
Gjermund Åsen
Tromsø
Bojan Zajić
Sarpsborg 08
Aron Elı́s Trándarson
Aalesund
Daniel Fredheim Holm Vålerenga
Eirik Hestad
Molde
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Minutes
2653
1286
2383
1000
1891
1709
1760
1152
1974
941
Total
0.8587
0.7967
0.7618
0.7167
0.6806
0.6745
0.6574
0.6464
0.6449
0.6258
0.7064
Shot
0.2702
0.2872
0.2373
0.2097
0.2128
0.1602
0.2360
0.2980
0.1309
0.0661
0.2108
Pass
0.3297
0.1726
0.2650
0.1888
0.2292
0.1151
0.2166
0.1567
0.2683
0.2771
0.2219
Take on
0.0242
0.0670
0.0449
0.0051
0.0198
0.0125
0.0304
0.0286
0.0439
0.0212
0.0298
Long pass
0.0253
0.0585
0.0048
0.0689
0.0071
0.0021
0.0140
0.0163
0.0124
0.0201
0.0229
Corner
0.0004
0.0119
0.0036
0.0703
0.0009
0.1190
0.0221
0.0095
0.0110
0.0846
0.0333
Ball carry
0.1227
0.0929
0.1352
0.0817
0.1100
0.0678
0.0550
0.0726
0.0978
0.0704
0.0906
on loan from a foreign club in 2015
Now consider the top 10 central midfielders shown in Table 5.18. Mike Jensen was the best
central midfielder in 2015 according to Model 2 by a wide margin, in sharp contrast to in Model
1 where he did not even make the top 10 list. This is a good example of how the difference
in rewarding shots has big implications for the lists. In fact, this is also evident for second and
third placed Etzaz Hussain and Harmeet Singh, who also were absent on the top 10 list in Model
1. Jensen had the highest impact value in shots, crosses and ball carries. A closer look on his
impact values both from Model 1 and Model 2 is given in Section 5.4.1. Third placed Harmeet
Singh had a low impact value from shots, and the highest from both passes and long passes
among the players in the table. This can indicate that he is a good passing player, both short and
long. Ole Kristian Selnæs is another example of this, and his impact values show above average
passing abilities in addition to crosses and corner kicks.
As for the list from Model 1, both Malaury Martin and Giorgi Gorozia features among
the central midfielders. The former was considered a big talent and captained every France
national youth team from U-17s through U-21s. The latter is an example of a player that was
not appreciated in the journalist ratings, with a rating of 99th and 101st in VG (2015) and
Altomfotball (2015), respectively. This makes him a candidate of unappreciated talent, and
considering he was born in 1995 he might be a player to watch in the future. Martin did not
play enough to obtain a rating in neither VG nor Altomfotball. In total, six of the players in the
top 10 list for central midfielders also appeared in the list from Model 1.
60
5.3 Markov Game Model 2
Table 5.18: Model 2: Top 10 central midfielders, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Team
Mike Jensen
Rosenborg
Etzaz Hussain*
Molde
Harmeet Singh
Molde
Giorgi Gorozia
Stabæk
Ole Kristian Selnæs*
Rosenborg
Fredrik Midtsjø
Rosenborg
Jone Samuelsen
Odd
Malaury Martin
Lillestrøm
Christian Grindheim
Vålerenga
Bismark Adjei-Boateng** Strømsgodset
Top 10 average
*
sold to foreign club during or after the 2015 season
**
Minutes
2466
1956
2212
1467
1951
2396
2292
975
2675
1293
Total
1.0463
0.7513
0.7429
0.6988
0.6865
0.6609
0.6473
0.6427
0.6139
0.5535
0.7044
Shot
0.2562
0.1343
0.0985
0.1110
0.0757
0.1205
0.1435
0.1252
0.0649
0.1999
0.1330
Pass
0.3341
0.3162
0.3671
0.2833
0.3080
0.3048
0.2440
0.2152
0.2832
0.2271
0.2883
Cross
0.0737
0.0099
0.0203
0.0304
0.0319
0.0143
0.0476
0.0290
0.0138
0.0348
0.0306
Long pass
0.0285
0.0458
0.0847
0.0205
0.0675
0.0246
0.0114
0.0442
0.0285
0.0149
0.0371
Corner
0.0929
0.0275
0.0128
0.0891
0.0511
0.0024
0.1375
0.0600
0.0007
0.0474
Ball carry
0.1235
0.1208
0.0946
0.0612
0.0875
0.0885
0.0960
0.0372
0.0567
0.0433
0.0809
on loan from a foreign club in 2015
A general observation is that the impact values from shots are decreasing with respect to
playing position on the field. The same can not be said about other values such as for passes
which seem more stable across playing positions. As players further back on the field are
considered, a larger portion of their values was due to passes than for the more offensive players.
Table 5.19 shows the top 10 list for full backs. Per-Egil Flo is on top of the list, and had high
impact values in passes, corner kicks and crosses but none of which are significantly higher than
for the rest. This can show that he had good overall performance in 2015, and Model 2 is able
to identify that. In fact, Flo was considered the best full back in both 2014 (see Table C.2) and
2015 by Model 1, in addition to being second placed in 2014 (see Table D.2) by Model 2, just
shy of Lars-Christopher Vilsvik.
Table 5.19: Model 2: Top 10 full backs, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Per-Egil Flo
Jonas Svensson
Espen Ruud
Lars-Christopher Vilsvik
Martin Linnes*
Joachim Olsen Solberg
André Danielsen
Birger Meling
Jørgen Skjelvik
Akeem Latifu*
Top 10 average
*
Team
Molde
Rosenborg
Odd
Strømsgodset
Molde
Mjøndalen
Viking
Stabæk
Rosenborg
Aalesund
sold to foreign club during or after the 2015 season
**
Minutes
1998
2610
2476
2126
2439
2486
2700
2263
1812
2564
Total
0.7505
0.7108
0.6584
0.6179
0.5079
0.4978
0.4634
0.4592
0.4468
0.4454
0.5558
Shot
0.0849
0.1126
0.0881
0.1047
0.0545
0.0895
0.0612
0.1000
0.0762
0.0657
0.0837
Pass
0.2954
0.3389
0.1954
0.2076
0.1931
0.0798
0.1425
0.1665
0.1873
0.1206
0.1927
Cross
0.1053
0.0730
0.1233
0.1542
0.0908
0.0517
0.0892
0.0279
0.0587
0.1373
0.0911
Corner
0.1187
0.0518
0.0237
0.0071
0.1017
0.0351
0.0484
0.0030
0.0389
Throw-in
0.0333
0.0412
0.0496
0.0371
0.0459
0.0222
0.0383
0.0301
0.0256
0.0363
0.0360
Ball carry
0.0470
0.0709
0.0471
0.0593
0.0430
0.0208
0.0523
0.0585
0.0620
0.0558
0.0517
on loan from a foreign club in 2015
Second placed Jonas Svensson was rated top 10 by both VG and Altomfotball, but was not
on the list for Model 1 due to his numerous missed shots. He has the highest value in the table
for both passes and ball carries. In addition, it seems that he is good at finding teammates on
throw-ins, which are often executed by the full backs. Lars-Christoper Vilsvik is the full back
with the highest impact value from crosses, and he also has a high shot value. Amongst the
top four full backs are the top three from Model 1, and in total six out of ten are the same. In
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Chapter 5. Evaluating Results
general, it can be seen from the impact value of crosses in the table, that the model is able to
value crosses as an important attribute for the full backs.
Finally, consider the top 10 list for the centre backs in Table 5.20, where six names are familiar from Model 1. As for Model 1, Johan Bjørdal is the highest rated centre back mostly due to
his high impact value from passing. He had below average impact values for aerial duels, which
is considered important for a player in his position. However, he had above average values for
both long passes and ball carries. Stefan Strandberg and Hólmar Örn Eyjólfsson, number two
and three on the list respectively, have roughly the same impact values in all categories, but
the former seems to have hit better long passes in 2015. All top three centre backs played for
Rosenborg in 2015. It is likely that Rosenborg had a higher than average possession, which can
lead to the centre backs seeing a lot of the ball and influence their impact value from passing
and ball carries positively. Next on the list, Rhett Bernstein has the highest impact value from
shots and aerial duels, who constitute a large part of his total impact value.
Table 5.20: Model 2: Top 10 centre backs, Tippeligaen 2015.
Average total value = 0.4608, Minimum total value = 0.0059
Player
Johan Bjørdal
Stefan Strandberg*
Hólmar Örn Eyjólfsson
Rhett Bernstein*
Joona Toivio
Jørgen Horn*
Oddbjørn Lie
Morten Sundli
Vegard Forren
Andreas Nordvik*
Top 10 average
*
Team
Rosenborg
Rosenborg
Rosenborg
Mjøndalen
Molde
Strømsgodset
Aalesund
Mjøndalen
Molde
Sarpsborg 08
sold to foreign club during or after the 2015 season
**
Minutes
1093
1151
2357
1234
1706
1260
1549
2018
2416
1846
Total
0.4473
0.4149
0.3758
0.3516
0.3293
0.3004
0.2715
0.2690
0.2614
0.2366
0.3258
Shot
0.1057
0.1064
0.0974
0.2193
0.1184
0.0737
0.0495
0.1332
0.0402
0.0830
0.1027
Pass
0.2274
0.1797
0.1835
0.1001
0.1363
0.1623
0.1161
0.0907
0.1300
0.0950
0.1421
Aerial
0.0002
0.0215
0.0124
0.0650
0.0107
-0.0008
-0.0024
0.0136
0.0067
0.0037
0.0131
Cross
0.0018
0.0057
0.0005
0.0034
0.0193
0.0125
0.0381
0.0106
0.0123
0.0072
0.0111
Long pass
0.0490
0.0613
0.0260
0.0046
0.0259
0.0335
0.0198
0.0180
0.0700
0.0630
0.0317
Ball carry
0.0803
0.0883
0.0742
0.0067
0.0290
0.0423
0.0311
0.0205
0.0353
0.0229
0.0430
on loan from a foreign club in 2015
For the players in defence (full backs and centre backs), few of the actions that measure
defensive contributions, like tackles, interceptions and ball recoveries are present in the tables.
This is because these actions have too low values to become a significant contributor to the
total impact value. This might indicate that Model 2 has limited ability to appreciate defensive
involvements.
As for Model 1, a league table can be obtained by aggregating the values for each team.
The obtained tables for Model 2 can be found in Appendix D, Tables D.3 and D.6 for 2014 and
2015 respectively. The table for 2014 suggests that Brann substantially underachieved during
the season according to Model 2, and they did possibly not deserve to have been in the relegation
battle. Once again, it can be seen from Table D.6 that Molde was inefficient in the 2015 season.
A closer look on the team evaluations from both Model 1 and Model 2 is found in Appendix E.
5.4
Case Studies
For a more in-depth description of how the models evaluate players, two of the most interesting
cases are investigated further. Mike Jensen was valued very differently across the two models
and seasons. Was he actually as good as people believe? Martin Ødegaard comes out as the best
player in his position in both models (see Appendixes C and D) and is considered one of the
62
5.4 Case Studies
most talented young footballers in the world (see for instance Mirror (2015), Gazetta (2014)).
He was wanted by many of the largest clubs in Europe after his performance in Tippeligaen
2014, and ended up signing for Spanish top club Real Madrid. What did he do better than the
rest of the players?
5.4.1
Mike Jensen
Figure 5.11 shows a box plot of some of the different actions performed by Mike Jensen from
Model 1, in order to examine his performance more accurately. The shots are excluded from
the figure, due to being an order of magnitude larger than the other involvements. A similar box
plot including the shots can be seen in Figure F.4 in Appendix F.1. In these figures, different
types of actions are on the x-axis and the impact value is on the y-axis. The box plots illustrate
the variation in each sample of data (that is, the variation in the impact by each type of action),
without making any assumption of the statistical distribution of the sample.
In a box plot, the blue boxes contain 50% of the observations, where the upper and lower
edges are the 75th percentile and the 25th percentile, respectively. The average value is in the
middle of the boxes, while the median is marked as a red line. Red crosses are observations
that are characterised as outliers, lying outside the whiskers which illustrate the 0.35th and
the 99.65th percentile. Practically, an outlier illustrate either a very good or poor involvement
by the player, compared with his average impact of performing this action. Box plots, with
and without shots, for Mike Jensen in 2014 are shown in Appendix F.1. Table 5.21 shows the
impact value per 90 minutes from Model 1 for the same actions. These values are calculated by
aggregating all the observations (which are illustrated in the figure) for each action and further
normalise them to a per 90 minutes basis.
Table 5.21: Model 1:
Impact values for Mike
Jensen 2015
Action
Pass
Take on
Foul won
Corner won
Tackle
Interception
Clearance
Shots
Aerial
Ball rec.
Cross
Long
Corner
Ball carry
Free kick
Total
Impact
0.0750
0.0168
0.0120
0.0110
0.0060
0.0017
0.0010
-0.1608
-0.0026
0.0073
0.0365
0.0082
0.0370
0.0257
0.0062
0.0783
Figure 5.11: Model 1: Box plot for Mike Jensen in 2015 excluding shots
As mentioned above, Mike Jensen was not appreciated by Model 1 as a top 10 player in
his position in 2015, although pundits seemed to agree he was one of the best of all players.
63
Chapter 5. Evaluating Results
In 2014, as can be seen from Table C.2, he was rated fifth in his position. In addition, Model
2 rates him as the best central midfielder in 2015, and the fourth best in 2014 (see Table D.2).
Why is this the case?
In Table 5.21, it can be seen that the total impact per 90 minutes from the Mike Jensens
shots is equal to -0.1608, which heavily influences the total impact value. Nine out of the top
10 central midfielders in Model 1 had a shot impact larger than zero. If the impact from his
shots instead was equal to zero, his total impact value would have been 0.2391, which would
have lead to Jensen being ranked as the third best central midfielder. Hence, this is a very good
example of how the shots influence the total impact value in Model 1.
The influence from shots is different in Model 2, where the player receives an impact equal
to the likelihood of scoring in the given state. Table 5.22 shows the impact value per 90 minutes
of the same actions for Model 2, while Figure 5.12 illustrates a box plot of these actions. In
this figure, it can be seen that the impact value from a shot always is greater than zero, and
significantly larger compared with the other actions. Hence, the total impact from shots is
positive for Jensen, as well as for most of the other players. The magnitude of the values in
Table 5.22 can not be directly compared with the values from Model 1 in Table 5.21. Instead,
the values of the different actions should be compared internally within each table, and with the
total impact value per 90 minutes shown at the bottom of each table.
Table 5.22: Model 2:
Impact values for Mike
Jensen 2015
Action
Pass
Take on
Foul won
Corner won
Tackle
Interception
Clearance
Shots
Aerial
Ball rec.
Cross
Long
Corner
Ball carry
Free kick
Total
Impact
0.3341
0.0352
0.0204
0.0301
0.0044
0.0010
-0.0042
0.2562
0.0031
0.0258
0.0737
0.0285
0.0929
0.1235
0.0141
1.0463
Figure 5.12: Model 2: Box plot for Mike Jensen in 2015
When comparing the plots and tables from the two models, some hypotheses regarding the
abilities of Mike Jensen can be made. It seems that he is a good passing player, which can be
seen from the large number of positive outliers in Figure 5.12, in addition to the high values in
Tables 5.21 and 5.22. Furthermore, it seems that he is a good carrier of the ball, in addition to
be able to take on opposing players. The figures and tables also indicate that Mike Jensen is
a good crosser and hits accurate corners. The total impact per 90 minutes from these actions
are positive in both models, although the median of both actions are below zero in Figure 5.11.
However, this is not surprising since both corners and crosses are usually cleared away by the
defenders.
64
5.4 Case Studies
In order to evaluate these hypotheses, the samples of values for Jensen in these actions can
be compared with the samples containing values for all players in Tippeligaen 2015. Figure
5.13 shows a comparison between Jensen and all players in Tippeligaen from Model 2 for the
2015 season without the outliers. The outliers are excluded because of their magnitude in the
samples for Tippeligaen as a whole. Since Jensen is a central midfielder, some of the more
defensive attributes are also included in the comparison. Amongst the more defensive actions,
it especially seems that Jensen is good at ball recoveries from the two tables.
Figure 5.13: Model 2: Comparison of Mike Jensen vs league for 2015
In all five offensive categories except for corners the medians of Jensen lie higher than the
median for Tippeligaen. Furthermore, the top of the box and the upper whisker are located
higher for Jensen in all five offensive categories. Hence, from the figure it seems that Mike
Jensen is better compared with the league as a whole in the following five categories: passing,
take ons, crosses, corners and ball carries. Jensen primarily plays to the right in a central
midfield of three players, where the player in the middle has more defensive duties. This allows
Jensen to support the attackers and contribute offensively. Based on this, the five categories
above can be considered as important capabilities for a player in the role of Jensen. In addition,
if the values for these actions from the two models, in Table 5.21 and 5.22, are aggregated, it
can be seen that they constitute a major part of the total impact by Jensen.
In addition to his offensive capabilities, Jensenis also known as a high-intensity player who
does not back down from a duel. From Figure 5.13 it can be seen that he is also an effective
player in some of the more defensive parts of the game. His median lies above the median of
Tippeligaen in both tackling, interceptions and ball recoveries. Furthermore, his upper whiskers
are also located a bit higher than for Tippeligaen as a whole. As previously indicated, it seems
that his impact from especially ball recoveries is significant. Hence, Jensen can also be considered a good duel player that wins the ball in favourable positions.
Overall, this case study has shown that Mike Jensen is a very good player in Tippeligaen. He
65
Chapter 5. Evaluating Results
has shown good abilities, especially offensively, but also defensively to some extent. However,
his willingness to take a lot of shots in 2015 lead to a total impact value per 90 minutes slightly
below average in Model 1. On the other hand, Model 2 rates him as the best central midfielder
and fourth overall among the players with over 900 minutes played in 2015.
66
5.4 Case Studies
5.4.2
Martin Ødegaard
To reveal what wonderkid Martin Ødegaard did better than the rest in Tippeligaen before completing a 35 million NOK transfer to Spanish club Real Madrid, an examination of the impact
made by Ødegaard in the 2014 season is done. Figures 5.14 and 5.15 show box plots containing
selected actions of Ødegaards involvements from Model 1 and 2 respectively. Tables 5.23 and
5.24 show the total impact value per 90 minutes from the same actions. A box plot from Model
1 including the impact from his shots can be seen in Figure F.5.
Table 5.23: Model 1:
Impact values for Martin
Ødegaard 2014
Action
Pass
Take on
Foul won
Corner won
Tackle
Interception
Clearance
Shot
Aerial
Ball rec.
Cross
Long
Corner
Ball carry
Free kick
Total
Impact
0.1340
0.0038
0.0130
0.0053
0.0083
-0.0008
-0.0002
0.1472
-0.0007
0.0100
0.0112
0.0068
0.0309
0.0154
-0.0052
0.4030
Figure 5.14: Model 1: Box plot for Martin Ødegaard in 2014 excluding
shots
Table 5.24: Model 2:
Impact values for Martin
Ødegaard 2014
Action
Pass
Take on
Foul won
Corner won
Tackle
Interception
Clearance
Shot
Aerial
Ball rec.
Cross
Long
Corner
Ball carry
Free kick
Total
Impact
0.5469
0.0594
0.0300
0.0201
0.0013
0.0002
-0.0022
0.1496
0.0008
0.0180
0.0117
0.0208
0.0552
0.2207
0,0045
1.1504
Figure 5.15: Model 2: Box plot for Martin Ødegaard 2014
67
Chapter 5. Evaluating Results
From the figures it seems that Ødegaard has positive involvements from many of the different actions. In one or both of the models all actions seem to have a positive impact, except interceptions, clearances and aerial duels. Surprisingly, Ødegaard playing as an attacking midfielder,
he seems to contribute positively through the more defensive actions tackles and ball recoveries.
Nonetheless, his contribution through interceptions and clearances seems to be more limited.
A closer examination of his actions, shows that Ødegaard only won 15 corners, attempted ten
crosses and took nine free kicks. The samples containing these actions are therefore considered
too small, despite showing positive impact values in one or both of the two tables. Furthermore,
one or both tables indicate that passes, take ons, shots, corners and ball carries are most influential among the actions. Especially the values for passing seem to heavily influence the total
impact value in both the models. In addition, the large value for ball carry in Model 2 is also
worth noting. Figure 5.16 shows a comparison of the impact made by Ødegaard and all players
in Tippeligaen within a selection of actions based on the above discussion.
Figure 5.16: Model 2: Comparison of Martin Ødegaard vs league for 2014
Based on the last box plot, it seems that Martin Ødegaard performed better compared to
the samples of all the players on at least six types of actions. His passing ability is good, he is
capable to take on a player, gets fouled in good positions, wins the ball in favourable positions,
can pick out a long pass and he is a good carrier of the ball. Martin Ødegaard plays primarily
as an attacking midfielder, and these abilities are believed to be important for such a player. In
addition, this is correspondence with the praise he has received for his passing and technical
abilities. Moreover, it seems like his defensive contributions are quite limited, but he had some
good tackles and interceptions. Further examinations revealed that he took 41 corners, which
seems to be a potential area of improvement for Ødegaard when looking at Figure 5.16, despite
having a positive impact on his total impact value for both models.
68
5.5 Reliability Out of Sample
5.5
Reliability Out of Sample
In order to further investigate the performance of the models, they have been tested out of
sample using data from the first 13 match days of the 2016 season. Reliability tests are done by
comparing the values of players who have played more than 390 minutes (corresponds to one
third of the season so far) in the 2016 season and over 900 minutes in 2015. Figure 5.17 shows
the results when these players are compared across seasons with respect to their G/xG value. In
Appendix G top lists for 2016 from the three models are presented.
Figure 5.17: xG Model: Scatter plot and correlation of G/xG between 2015 and 2016
As can be seen in Figure 5.17, the correlation coefficient between 2015 and 2016 is -0.0891.
This clearly indicates that the players do not manage to maintain their efficiencies across seasons, which was also seen when examining 2014 to 2015. However, the correlation coefficient
between 2015 and 2016 is 0.7565, when comparing the xG/90min values. Hence, the players
evaluated manage to maintain their respective xG/90min value across the two seasons. This
indication was also evident when evaluating in sample data.
Figures 5.18 and 5.19 show the results when the players are compared across seasons based
on their average impact value per 90 minutes from the two Markov models.
The scatter plot in Figure 5.18 illustrates the values from Model 1 for the two seasons.
From the plot it seems that the players does not manage to maintain their respective values
from 2015 to 2016, and the correlation coefficient is 0.0932, which is somewhat lower than in
sample. However, as previously pointed out, the values from shots heavily influence the value
of each player. As illustrated by the low correlation of G/xG, the players are often not capable
of maintaining their effectiveness over seasons, which can impact the value of a player in Model
1 negatively. Thus, the reliability of Model 1 seems to be limited due to the heavy influence by
the outcome of shots.
In Figure 5.19 the scatter plot of the player values from Model 2 is shown. The correlation
coefficient is as high as 0.9006, which is almost exactly equal to the coefficient when comparing
the 2014 and 2015 seasons. This indicates that Model 2 is reliable in assessing the impact on
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Chapter 5. Evaluating Results
Figure 5.18: Model 1: Scatter plot and correlation between 2015 and 2016
the field by each player, also when assessing out of sample data.
Figure 5.19: Model 2: Scatter plot and correlation between 2015 and 2016
70
5.6 Evaluation of Research Questions
5.6
Evaluation of Research Questions
In this section, answers to the three research questions, introduced in Section 1.2, are provided
and discussed on the basis of the previously presented results.
RQ 1 Is it possible to create a statistically significant xG model that assesses the quality of all
shots in Tippeligaen in order to evaluate the efficiency of primarily offensive players?
An xG model for Tippeligaen has been developed in this thesis. The results from the logistic
regression show that it is possible to create a model with several statistically significant in-match
variables that affect the likelihood of converting a shot into a goal. The explanatory variables
include multiple aspects that influence the likelihood of scoring, and their signs can be argued
for from a footballing point of view. The estimated xG/90 min values for individuals playing
more than a third of the season are fairly consistent across seasons for in-sample data, and the
xG/90 min also shows to be reliable on a reasonable level across seasons when testing on out of
sample data.
However, the xG Model has some important limitations. The one considered to be the most
prominent is the inability to model defender proximity. Defenders that cover attacking players
and block their shots are an important aspect that has been shown by other works to have an
impact on the likelihood of scoring. The data set utilised in this thesis does not support such
an analysis, which has to be a subject for further research. In addition, the free kick model
exhibited low discrimination, which also impacts the likelihood of the shots for some players.
Furthermore, the data set used in building the xG Model is also a source of error. Human
annotators are collecting the data, which gives rise to bias for some variables. In addition, some
variables are calculated by the authors, which introduces the possibility of miscalculation.
The player evaluations based on efficiency measured by G/xG is to some extent as what
should be expected. However, the G/xG performance measure exhibits poor reliability across
seasons. Thus, the xG Model assigns likelihoods that enables a G/xG player rating only for the
season in question. Due to low correlation coefficients across seasons both in and out of sample,
few inferences can be made on the consistency of the performance measure G/xG of players.
When assessing players across seasons it is more interesting to look at their xG/90 min
value. The xG/90 min assigned to each player has a higher reliability compared to G/xG across
seasons. The total xG/90 min is a product of the number of shots and the quality of the shot,
indicating that a player with high xG/90 min shoots a lot, creates high quality shots or is responsible for a favourable mix of these two factors. By using xG/90 min as the primary performance
measure, one can make more reliable inferences on player performance across seasons.
RQ 2 Is it possible to create a Markov game model for football that is able to evaluate all player
involvements in a match and rate players over the course of a season?
Two variations of a Markov game model applied to football have been developed in this thesis.
To the best of the authors knowledge, neither of the two approaches have been attempted on
data from football before. The results show that it is possible to create models that are able
to assign values to individual player involvements with such approaches. The Markov game
models also have the ability to investigate individual players and evaluate both offensive and
defensive contributions, and show promising results when comparing them to three benchmarks.
However, due to the nature of the game and the model specifications, the values of offensive
contributions are often orders of magnitude larger than the defensive. This has implications
71
Chapter 5. Evaluating Results
on the player ratings for the defensive positions, where the results show that they are primarily
rewarded for their offensive contributions and not, as would have been preferable, for their
defensive abilities.
The two models differ on some important features. The models are distinct in the way they
assign values to the different actions, which has some implications on the aggregated impact
value assigned to each player. This distinction is mainly due to different impact functions and
differences in features between the two models. This has especially two large consequences,
where the first is with regard to the impact value of shots. Model 1 has a bias towards players
with high efficiencies, due to the way the impact function is defined. It seems that most players
do not manage to maintain their efficiency rate across seasons, indicated by the low correlation coefficient of G/xG. This is believed to impact the reliability of Model 1, which shows a
correlation of 0.1944 between the two seasons. For Model 2, the impact function leads to a
favouring of players that regularly attempts high quality shots, because less emphasis is given
to the outcome. It seems that players manage to maintain their ability to generate high quality
shots across seasons to a larger extent compared with their efficiency. This is believed to positively affect the reliability of Model 2, which shows a correlation coefficient of 0.8988 between
the two seasons. Despite the large influence by shots in both models, they also seem to manage
to appreciate players that impact the game through other actions. When the correlations are
tested on out of sample data, the same result regarding their reliability can be observed. Model
2 shows a correlation of 0.9006, which is almost exactly the same as in sample, while Model 1
shows a mere 0.0932.
The second consequence from the different impact functions is regarding the evaluation of
the defensive involvements in football. Model 1 indicates a slightly larger appreciation for the
defensive involvements. This is observed from tackles, clearances and ball recoveries being
among the top five influential types of actions for one or more positions in Model 1, as seen
in Tables 5.10 to 5.12, while only aerial duels is visible among the defensive involvements in
Model 2. Since an action is valued on the basis of what is actually lead to in Model 1, it seems
that the model manage to occasionally recognise some good defensive involvements, which
leads to a more significant impact over the course of a season than for Model 2. For both
models, the offensive involvements are larger in magnitude than the defensive ones, which is
unsurprising due to them being closer to the objective of the game, namely scoring goals. This
leads to a somewhat unfair evaluation of the defensive players. As seen in the player tables in
Sections 5.2.3 and 5.3.3, the majority of the impact from the best defenders were from shots
and different passing events.
The results from the two models are somewhat different, although a total of 37 out of 60
players in the tables in Sections 5.2.3 and 5.3.3 are the same. An observation that can be made
regarding the difference between the two models, is that the impact values are higher for Model
2, which also shows better reliability. A suspicion that might be able to explain this, is that
fewer involvements receive a negative impact value in Model 2 than in Model 1. For instance,
a successful backward pass might more often be assigned a negative impact value in Model 1.
In Model 2, the value of an action is equal to the value of the state, which is defined as the
average values of the subsequent states. Thus, even though the pass was played backwards on
the field, it can receive a positive impact value if the average impact of the possible events that
followed was favourable. This can be seen when taking a closer look at the box plots from each
of the models for Mike Jensen, shown in Figure 5.11 and 5.12. When comparing the sample of
his passes, it is evident that the median in Model 1 is approximately zero, while it is slightly
positive in Model 2. Furthermore, the lower whisker is located lower for Model 1, in addition
72
5.6 Evaluation of Research Questions
to several negative outliers being evident. This indicates that the impact values from passes are
more evenly distributed around zero in Model 1, while a significantly larger share of the values
are positive in Model 2. With this in mind, it is reasonable to believe that the number of actions
and the total impact value are positively correlated for Model 2. However, when examining
these correlations for both the models, the results show that there is virtually no correlation
between the number of actions performed and the total impact value in neither of the models.
This difference between the models is believed to be more evident among the more offensive
involvements than the involvements further back on the field. When performing an action on
the offensive third of the field, it is likely that the possible subsequent involvements are positive
for the team in possession, which leads to a positive impact value for the action in Model 2. In
Model 1, the value of the action would depend on the action that follows next, which could be
an unfavourable action for the team in possession. Offensive involvements are therefore more
likely to be assigned positive values in Model 2, which is believed to result in a high number of
offensive involvements being favourable in order to obtain a high impact value. Furthermore,
players on the strongest teams are more likely to be in offensive positions more often, hence,
Model 2 might be more biased towards the players on the strongest teams. This can be supported
by the fact that many of the highest ranked players are from the teams considered the strongest.
In general, both models fulfil their purpose, which is to value each player involvement in
Tippeligaen. The results seem to be fairly reasonable, and the players that are considered the
best seem to be appreciated and identified as good players by the models. However, the models
seem to be very influenced by shots, as well as having a bias towards more offensive involvements. This is therefore considered as potential areas of improvements.
RQ 3 Is it possible to reveal undiscovered talent or identify under- and overvalued players
based on the evaluation of individual player involvements?
The results from the three models developed in this thesis show that it might be possible to
identify undiscovered talent. The results can be compared to other benchmarks, in order to
obtain an objective measure on the performance of a player relative to others.
For the Markov game models, a possible example is the 21 year old central midfielder
Giorgi Gorozia, who was rated third and fourth in his position in 2015 by Model 1 and Model 2,
respectively. The two subjective player ratings, used as benchmarks, rated him 99th and 101st
amongst all players, which indicate that he did not attract much attention from journalists and
pundits. He is currently valued at 300 thousand GBP by Transfermarkt. This is only slightly
higher than the current average in Tippeligaen of 270 thousand GBP, and substantially lower
than the average among the top 10 attacking midfielders of 828 thousand GBP. The players
above Gorozia on the list from Model 2, Mike Jensen, Etzaz Hussain and Harmeet Singh, are
valued at 1.35 million, 750 thousand and 825 thousand GBP, respectively. He was rated one
place above Ole Kristian Selnæs who is valued at 2.25 million GBP by Transfermarkt. All of
these players have estimated market values twice that of Gorozia, and could indicate that he is
an undervalued player compared to his colleagues.
In addition, although he might not be undiscovered, under- or overvalued, Martin Ødegaard
was the highest rated attacking midfielder in both Model 1 and Model 2 in 2014. This shows that
both models were able to rate a player that is considered by many as one of the most promising
young players in the world as the best.
Only the future can tell how the market value of a player will develop, and it is not possible
to conclude from the models in this thesis whether players actually are under- or overvalued.
73
Chapter 5. Evaluating Results
Nonetheless, as was a big motivation for this thesis, the models can open up new possibilities for
clubs both in measuring performance internally and for scouting players. The same detailed data
for every on-the-ball contact used in this thesis is collected during every match in 30 leagues
and competitions worldwide. Thus, the models developed in this thesis can have several areas
of application for clubs. They can use it internally to keep track of performance in own squads
throughout a season, to assess what attributes their players possess or can improve, as well as
to scout for talent either domestically or internationally.
Furthermore, clubs can use the two Markov game models to quantify what attributes each
player is good at compared to others, which can be valuable information when replacing players
in the different positions. The values in the state space learnt from data on Tippeligaen can be
used to identify players in foreign leagues that are able to obtain high impact values from what
is considered good by the model for Norwegian football. On the other hand, the values in the
state space can also be learnt on a data set from a foreign league to identify the best players in
that country. By doing this, clubs can obtain short lists of players to scout more closely, which
can make the process less time consuming and possibly more effective. Scouts can spend more
of their time on assessing qualitative considerations, which is still a major part of the process in
acquiring players on the transfer market.
For the xG Model, it is possible to follow the same approach. Either by running a binary
logistic regression on shots attempted in a foreign league or to use the xG Model built for
Tippeligaen, it might be possible to identify the most efficient players in converting shots to
goals. However, as argued for in the answer to RQ1, the performance measure G/xG shows
limited stability across seasons both in and out of sample, and it may not be wise to make
decisions based solely on player efficiency in one season. A possible strategy can be to look
for players that obtain a desirable xG/90 min and actually are able to replicate their efficiencies
across seasons, and scout them more closely.
74
Chapter
6
Conclusion
This thesis has described and documented the development of three models for evaluating individual player involvements in association football. To the authors knowledge, the xG Model
is the first of its kind for Norwegian football, and the two Markov game models are the first of
their kind in the entire football analytics community. The xG Model rated primarily offensive
players based on their efficiency in converting attempted shots to goals, while the two Markov
game models assigned values to all player involvements during a match, to identify what players had the biggest impact over the course of a season. Answers to the three research questions,
introduced in Chapter 1, has been given after the presentation and discussion of the results in
Chapter 5.
The results are deemed promising, although some remaining challenges and limitations are
identified. The xG model does not account for defender proximity, for which the literature
suggests plays an important role in such models. The framework for the Markov models, as
adopted from literature on ice hockey and basketball, seems to have a place in football as well.
However, the two Markov models show limited abilities in valuing defensive contributions,
which is addressed as a subject for further research. In addition, lack of comparable sources
make the reliability and validity of the models hard to assess.
Similar data as used in this thesis is available from 30 leagues and competitions worldwide.
Thus the three models developed can have several areas of application for clubs. For the xG
Model, it is possible to use a similar model building approach to identify the most efficient players in foreign leagues. Clubs can use the two Markov game models to quantify what attributes
each player is good at compared to others, which can be valuable information when replacing
players in the different positions. All of the three models can also be used to identify players
in foreign leagues in order to make short lists of players to scout more closely, which can make
the process less time consuming and possibly more effective. By doing this, scouts can spend
more of their time focusing on qualitative considerations.
Competition is fierce in the world of football. Only small improvements in scouting, squad
management and performance monitoring can have a huge monetary impact considering the
money in circulation. Getting it right more often than not can lead to a competitive advantage
in the future, and can be the difference between success and failure in the long run. Albeit
not flawless, the authors believe the models developed in this thesis can serve as a valuable
contribution to Rosenborg Ballklub and the football analytics community, especially with regard
to Norwegian football, which has not been studied widely earlier.
75
Chapter 6. Conclusion
76
Chapter
7
Recommendations for Further Research
The three models developed in this thesis are, to the best of the authors knowledge, the first of
their kind for Norwegian football. In the academic community, only one other xG model on
football is identified (Lucey et al. (2014)), while three articles were found that applies Markov
models to football, all by the same authors (Hirotsu and Wright (2002), Hirotsu and Wright
(2003a), Hirotsu and Wright (2003b)). However, no Markov model applied for assessing individual players was found. The novelty of the approaches in this thesis therefore make the
recommendations for further research important to address.
As mentioned earlier, the data used in this thesis is not spatiotemporal. Working with eventbased data from Opta makes it impossible to account for defender proximity. This is a variable,
which in other leagues is found to influence the likelihood of shot outcomes, and incorporating
defender proximity would be an important extension to the xG Model developed in this thesis.
Applying spatiotemporal data for the Markov models is also an important approach to consider
for future research. This would increase the detail level, possibly enabling a more accurate
calculation of player impact. Applying spatiotemporal data would therefore be an important
step towards developing models that are able to describe more of the complexity of football.
Furthermore, with access to more data it can be possible to include action histories in the
definition of a state, similar to the Markov game model for ice hockey presented in Routley
(2015). By utilising such an extension, historical effects from the game in a given state would
be accounted for, possibly influencing the value of the actions. As described, the two Markov
models share a bias towards shots and offensive involvements, although in somewhat different ways. A natural extension, and an important improvement for rating defensive players, is
finding a solution for putting less emphasis on offensive events and model involvements further
back on the field adequately. Including action histories might be able to appreciate a defensive
involvement to a larger extent if the preceding actions were performed by the opposing team.
In addition, a more comprehensive data set would also likely improve the accuracy of the xG
model. It would ensure a better basis for regular shot, but especially for static events it would
be possible to create models with better discrimination abilities. Situational variables like the
home field advantage, team strength and match status are shown to have an impact on match
outcome, and are accounted for in this thesis. With more data on hand, it might be possible to
incorporate other situational variables, like playing on artificial turf or the fatigue of players.
This could be important improvements for both the Markov models and the xG Model.
Another possible extension of the Markov models, is to analyse teams more thoroughly.
Especially with more data at hand, it might be possible to examine each team in detail, in order
77
Chapter 7. Recommendations for Further Research
to reveal patterns in their playing style. This could be useful for clubs when scouting their upcoming opponent. Such analyses is believed to especially provide insight if spatiotemporal data
is used. Furthermore, if such patterns are possible to reveal, it can be used to simulate matches
between the different teams, which could be useful in prediction and for betting companies.
In the two Markov models presented in this thesis, the total player impact is divided into
several, equally weighted categories. A possible extension is to create player ratings for specific
positions, which can be based on a weighted average of selected actions that are considered important for the different positions. Another interesting approach for utilising different categories
of player performance is to find determinants of the market value. Using a set of chosen performance categories, along with other variables like player age, contract duration and nationality,
one can specify a regression on market value from Transfermarkt. This might provide insight on
which specific capabilities that actually affects the market value of players in different positions,
which could be useful for clubs.
78
Bibliography
Alcock, A., 2010. Analysis of direct free kicks in the women’s football World Cup 2007. European Journal of Sport Science 10 (4), 279–284.
Almeida, C. H., Ferreira, A. P., Volossovitch, A., 2014. Effects of match location, match status
and quality of opposition on regaining possession in UEFA Champions League. Journal of
human kinetics 41 (1), 203–214.
Altomfotball, 2014. http://www.altomfotball.no/elementsCommonAjax.
do?cmd=statistics&subCmd=playerScore&tournamentId=1&seasonId=
336&teamId=&useFullUrl=false, Online; accessed 20-April-2016.
Altomfotball, 2015. http://www.altomfotball.no/elementsCommonAjax.
do?cmd=statistics&subCmd=playerScore&tournamentId=1&seasonId=
337&teamId=&useFullUrl=false, Online; accessed 20-April-2016.
Amir, E., Livne, G., 2005. Accounting, valuation and duration of football player contracts.
Journal of Business Finance & Accounting 32 (3-4), 549–586.
Anderson, C., Sally, D., 2013. The Numbers Game: Why Everything You Know About Soccer
Is Wrong, 1st Edition. Penguin Books, City of Westminster, London.
Bakkerode, M., Koning, R., 2014. The conversion of penalty kicks in the English Premier
League. Master’s thesis at Rijksuniversiteit Groningen.
Barreira, D., Garganta, J., Guimarães, P., Machado, J. C., Anguera, M. T., 2014. Ball recovery
patterns as a performance indicator in elite soccer. Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology 228 (1), 61–72.
Barreira, D., Garganta, J., Pinto, T., Valente, J., Anguera, M., Nunome, H., Drust, B., Dawson,
B., 2013. Do attacking game patterns differ between first and second halves of soccer matches
in the 2010 FIFA World Cup? In: Science and football VII: the proceedings of the seventh
world congress on science and football. London and New York: Routledge. pp. 193–198.
Bialkowski, A., Lucey, P., Carr, P., Yue, Y., Matthews, I., 2014a. ”win at home and draw away”:
automatic formation analysis highlighting the differences in home and away team behaviors. Proceedings of the 2014 MIT Sloan Sports Analytics Conference, http://www. sloansportsconference.com.
79
Bialkowski, A., Lucey, P., Carr, P., Yue, Y., Sridharan, S., Matthews, I., 2014b. Identifying team
style in soccer using formations learned from spatiotemporal tracking data. In: Data Mining
Workshop (ICDMW), 2014 IEEE International Conference on. IEEE, pp. 9–14.
Bialkowski, A., Lucey, P., Carr, P., Yue, Y., Sridharan, S., Matthews, I., 2014c. Large-scale
analysis of soccer matches using spatiotemporal tracking data. In: Data Mining (ICDM),
2014 IEEE International Conference on. IEEE, pp. 725–730.
Bjertnes, V., Nørstebø, O., Vabo, E., 2015. Identifying key performance indicators in Norwegian
Association Football. Project report at Norwegian University of Science and Technology.
Bojinov, I., Bornn, L., 2016. The Pressing Game: Optimal Defensive Disruption in Soccer.
In: Proceedings of the 2016 MIT Sloan Sports Analytics Conference, http://www. sloansportsconference.com.
Bradley, P. S., Lago-Peñas, C., Rey, E., Gomez Diaz, A., 2013. The effect of high and low
percentage ball possession on physical and technical profiles in English FA Premier League
soccer matches. Journal of Sports Sciences 31 (12), 1261–1270.
Bray, K., Kerwin, D., 2003. Modelling the flight of a soccer ball in a direct free kick. Journal of
sports sciences 21 (2), 75–85.
Bray, S. R., Law, J., Foyle, J., 2003. Team quality and game location effects in English professional soccer. Journal of Sport Behavior 26 (4), 319.
Brillinger, D. R., 2007. Modelling some norwegian soccer data. Advances in Statistical Modeling and Inference 3, 3–20.
Brown Jr, T. D., Van Raalte, J. L., Brewer, B. W., Winter, C. R., et al., 2002. World Cup soccer
home advantage. Journal of Sport Behavior 25 (2), 134.
Buraimo, B., Forrest, D., Simmons, R., 2010. The 12th man?: refereeing bias in English and
German soccer. Journal of the Royal Statistical Society: Series A (Statistics in Society)
173 (2), 431–449.
Buraimo, B., Simmons, R., Maciaszczyk, M., 2012. Favoritism and referee bias in European
soccer: evidence from the Spanish league and the UEFA Champions League. Contemporary
Economic Policy 30 (3), 329–343.
Carmichael, F., Thomas, D., 2005. Home-field effect and team performance evidence from
English Premiership football. Journal of sports economics 6 (3), 264–281.
Castellano, J., Casamichana, D., Lago, C., 2012. The use of match statistics that discriminate
between successful and unsuccessful soccer teams. Journal of human kinetics 31, 137–147.
Cervone, D., D’Amour, A., Bornn, L., Goldsberry, K., 2014. Pointwise: Predicting points and
valuing decisions in real time with nba optical tracking data. In: Proceedings of the 2014
MIT Sloan Sports Analytics Conference, http://www. sloansportsconference.com. Vol. 28.
Chang, Y.-H., Maheswaran, R., Su, J., Kwok, S., Levy, T., Wexler, A., Squire, K., 2014. Quantifying shot quality in the NBA. In: Proceedings of the 2014 MIT Sloan Sports Analytics
Conference, http://www. sloansportsconference.com.
80
Clarke, S. R., Norman, J. M., 1995. Home ground advantage of individual clubs in English
soccer. The Statistician 44 (4), 509–521.
Coleman, B. J., 2012. Identifying the “players” in sports analytics research. Interfaces 42 (2),
109–118.
Collet, C., 2013. The possession game? A comparative analysis of ball retention and team
success in European and international football, 2007–2010. Journal of Sports Sciences 31 (2),
123–136.
Constantinou, A. C., Fenton, N. E., Pollock, L. J. H., 2014. Bayesian networks for unbiased
assessment of referee bias in association football. Psychology of Sport and Exercise 15 (5),
538–547.
De Baranda, P. S., Lopez-Riquelme, D., 2012. Analysis of corner kicks in relation to match
status in the 2006 World Cup. European Journal of Sport Science 12 (2), 121–129.
Duarte, R., Araújo, D., Folgado, H., Esteves, P., Marques, P., Davids, K., 2013. Capturing
complex, non-linear team behaviours during competitive football performance. Journal of
Systems Science and Complexity 26 (1), 62–72.
Duarte, R., Araújo, D., Freire, L., Folgado, H., Fernandes, O., Davids, K., 2012. Intra-and intergroup coordination patterns reveal collective behaviors of football players near the scoring
zone. Human Movement Science 31 (6), 1639–1651.
Fearnhead, P., Taylor, B. M., 2010. On estimating the ability of NBA players. Journal of quantitative analysis in sports 7 (3), Article 11.
Fernando, T., Wei, X., Fookes, C., Sridharan, S., Lucey, P., 2015. Discovering methods of
scoring in soccer using tracking data. In: Proceedings of 2015 KDD Workshop on LargeScale Sports Analytics, http://www.large-scale-sports-analytics.org/.
Gazetta,
2014.
http://www.gazzettaworld.com/features/
greatest-young-players/5/, Online; accessed 24-May-2016.
Goumas, C., 2014. Home advantage in Australian soccer. Journal of Science and Medicine in
Sport 17 (1), 119–123.
Groebner, D., Shannon, P., Fry, P., Smith, K., 2011. Business Statistics: A Decision-Making
Approach, 8th Edition. Prentice Hall/Pearson, Upper Saddle River, N.J.
Gudmundsson, J., Wolle, T., 2014. Football analysis using spatio-temporal tools. Computers,
Environment and Urban Systems 47, 16–27.
Gyarmati, L., Hefeeda, M., 2016. Analyzing in-game movements of soccer players at scale
abs/1603.05583.
URL http://arxiv.org/abs/1603.05583
Gyarmati, L., Kwak, H., Rodriguez, P., 2014. Searching for a unique style in soccer. In: Proceedings of 2014 KDD Workshop on Large-Scale Sports Analytics, http://www.large-scalesports-analytics.org/.
81
Halvorsen, P., Sægrov, S., Mortensen, A., Kristensen, D. K., Eichhorn, A., Stenhaug, M., Dahl,
S., Stensland, H. K., Gaddam, V. R., Griwodz, C., et al., 2013. Bagadus: an integrated system
for arena sports analytics: a soccer case study. In: Proceedings of the 4th ACM Multimedia
Systems Conference. ACM, pp. 48–59.
Hirotsu, N., Wright, M., 2002. Using a markov process model of an association football match
to determine the optimal timing of substitutions and tactical decisions. Journal of the Operational Research Society 53 (1), 88–96.
Hirotsu, N., Wright, M., 2003a. Determining the best strategy for changing the configuration of
a football team. Journal of the Operational Research Society 54 (1), 878–887.
Hirotsu, N., Wright, M., 2003b. An evaluation of characteristics of teams in association football
by using a markov process model. Journal of the Royal Statistical Society: Series D (The
Statistician) 52 (4), 591–602.
Hosmer, D. W., Lemeshow, S., 2000. Applied Logistic Regression, 2nd Edition. John Wiley &
Sons, 605 Third Avenue, New York.
Hughes, M., Franks, I., 2005. Analysis of passing sequences, shots and goals in soccer. Journal
of Sports Sciences 23 (5), 509–514.
Hvattum, L. M., Arntzen, H., 2010. Using ELO ratings for match result prediction in association
football. International Journal of Forecasting 26 (3), 460–470.
Jankovic, A., Leontijevic, B., Pasic, M., Jelusic, V., 2011. Influence of certain tactical attacking
patterns on the result achieved by the teams participants of the 2010 FIFA World Cup in
South Africa. Physical Culture 65 (1), 34–45.
Jordet, G., Hartman, E., 2008. Avoidance motivation and choking under pressure in soccer
penalty shootouts. Journal of Sports & Exercise Psychology 30, 450–457.
Kerr, M. G. S., 2015. Applying machine learning to event data in soccer. Master’s thesis at
Massachusets Institute of Technology.
Kim, H.-C., Kwon, O., Li, K.-J., 2011. Spatial and spatiotemporal analysis of soccer. In: Proceedings of the 19th ACM SIGSPATIAL international conference on advances in geographic
information systems. ACM, pp. 385–388.
Kumar, V., Kaur, M., Sharma, D. P., 2013. Effect of speed endurance training on crossing
accuracy in soccer. International Journal of Movement Education and Sports Sciences 2, 99–
104.
Lago, C., Casais, L., Dominguez, E., Sampaio, J., 2010. The effects of situational variables on
distance covered at various speeds in elite soccer. European Journal of Sport Science 10 (2),
103–109.
Lago, C., Martı́n, R., 2007. Determinants of possession of the ball in soccer. Journal of Sports
Sciences 25 (9), 969–974.
Lago-Ballesteros, J., Lago-Peñas, C., 2010. Performance in team sports: identifying the keys to
success in soccer. Journal of Human Kinetics 25, 85–91.
82
Lago-Peñas, C., 2012. The role of situational variables in analysing physical performance in
soccer. Journal of human kinetics 35 (1), 89–95.
Lago-Peñas, C., Dellal, A., 2010. Ball possession strategies in elite soccer according to the evolution of the match-score: the influence of situational variables. Journal of Human Kinetics
25, 93–100.
Littman, M. L., 2001. Value-function reinforcement learning in Markov games. Journal of Cognitive Systems Research (2), 55–66.
Lopes, J., Araújo, D., Peres, R., Davids, K., Barreiros, J., 2008. The dynamics of decision
making in penalty kick situations in association football. The Open Sports Sciences Journal
1 (1), 24–30.
Lucey, P., Bialkowski, A., Monfort, M., Carr, P., Matthews, I., 2014. quality vs quantity: Improved shot prediction in soccer using strategic features from spatiotemporal data. MIT Sloan
Sports Analytics Conference.
Lucey, P., Oliver, D., Carr, P., Roth, J., Matthews, I., 2013a. Assessing team strategy using
spatiotemporal data. In: Proceedings of the 19th ACM SIGKDD international conference on
Knowledge discovery and data mining. ACM, pp. 1366–1374.
Lucey, P., Oliver, D., Carr, P., Roth, J., Matthews, I., 2013b. Assessing team strategy using
spatiotemporal data. In: Proceedings of the 19th ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining. KDD ’13. ACM, New York, NY, USA, pp. 1366–
1374.
URL http://doi.acm.org/10.1145/2487575.2488191
Macdonald, B., 2010. A regression-based adjusted plus-minus statistic for nhl players. Journal
of Quantitative Analysis in Sports 7 (3).
Macdonald, B., 2011a. Adjusted plus-minus for nhl players using ridge regression with goals,
shots, fenwick, and corsi. arXiv e-print arXiv:1201.317.
Macdonald, B., 2011b. An improved adjusted plus-minus statistic for NHL players. In: Proceedings of the MIT Sloan Sports Analytics Conference, URL http://www. sloansportsconference. com.
Macdonald, B., 2012. An expected goals model for evaluating nhl teams and players. In: Proceedings of the 2012 MIT Sloan Sports Analytics Conference, http://www. sloansportsconference. com.
Macdonald, B., Lennon, C., Sturdivant, R., 2012. Evaluating nhl goalies, skaters, and teams
using weighted shots. arXiv preprint arXiv:1205.1746.
Macdonald, B., Weld, C., Arney, D. C., 2013. Quantifying playmaking ability in hockey. arXiv
preprint arXiv:1307.6539.
Machado, J. C., Barreira, D., Garganta, J., 2014. The influence of match status on attacking
patterns of play in elite soccer teams. Revista Brasileira de Cineantropometria & Desempenho
Humano 16 (5), 545–554.
83
McHale, I., Scarf, P., 2007. Modelling soccer matches using bivariate discrete distributions with
general dependence structure. Statistica Neerlandica 61 (4), 432–445.
McHale, I. G., Scarf, P. A., Folker, D. E., 2012. On the development of a soccer player performance rating system for the english premier league. Interfaces 42 (4), 339–351.
Mechtel, M., Bäker, A., Brändle, T., Vetter, K., 2011. Red cards: not such bad news for penalized guest teams. Journal of Sports Economics 12, 621–646.
Mirror,
2015.
http://www.mirror.co.uk/sport/football/news/
20-most-promising-youngsters-world-6034191, Online; accessed 24May-2016.
Misirlisoy, E., Haggard, P., 2014. Asymmetric predictability and cognitive competition in football penalty shootouts. Current Biology 24 (16), 1918–1922.
Moura, F. A., Martins, L. E. B., Anido, R. D. O., De Barros, R. M. L., Cunha, S. A., 2012. Quantitative analysis of Brazilian football players’ organisation on the pitch. Sports Biomechanics
11 (1), 85–96.
Niu, Z., Gao, X., Tian, Q., 2012. Tactic analysis based on real-world ball trajectory in soccer
video. Pattern Recognition 45 (5), 1937–1947.
Noël, B., Furley, P., Van der Kamp, J., Dicks, M., Memmert, D., 2015. The development of
a method for identifying penalty kick strategies in association football. Journal of sports
sciences 33 (1), 1–10.
Opta, 2016. https://www.optasports.com, Online; accessed 24-January-2016.
Orth, D., Davids, K., Araújo, D., Renshaw, I., Passos, P., 2014. Effects of a defender on run-up
velocity and ball speed when crossing a football. European Journal of Sport Science 14 (1),
316–323.
Pettigrew, S., 2015. Assessing the offensive productivity of nhl players using in-game win probabilities. In: 9th Annual MIT Sloan Sports Analytics Conference. Vol. 2. p. 8.
Pollard, R., Silva, C., Medeiros, N., 2008. Home advantage in football in Brazil: differences between teams and the effects of distance traveled. Revista Brasileira de Futebol (The Brazilian
Journal of Soccer Science) 1 (1), 3–10.
Poole, D. L., Mackworth, A. K., 2010. Artificial Intelligence: foundations of computational
agents. Cambridge University Press, 32 Avenue of the Americas, New York, NY, USA.
Premier League, 2016. http://www.premierleague.com/en-gb/players/
ea-sports-player-performance-index.html, Online; accessed 19-May-2016.
Reep, C., Benjamin, B., 1968. Skill and chance in association football. Journal of the Royal
Statistical Society. Series A (General) 131 (4), 581–585.
Reep, C., Pollard, R., Benjamin, B., 1971. Skill and chance in ball games. Journal of the Royal
Statistical Society. Series A (General) 134 (4), 623–629.
84
Reilly, B., Witt, R., 2013. Red cards, referee home bias and social pressure: evidence from
English Premiership soccer. Applied Economics Letters 20 (7), 710–714.
Ridder, G., Cramer, J., Hopstaken, P., 1994. Down to ten: estimating the effect of a red card in
soccer. Journal of the American Statistical Association 89 (427), 1124–1127.
Routley, K. D., 2015. A Markov game model for valuing player actions in ice hockey. Master’s
thesis at Simon Fraser University.
Ruiz-Ruiz, C., Fradua, L., Fernández-GarcÍa, Á., Zubillaga, A., 2013. Analysis of entries into
the penalty area as a performance indicator in soccer. European Journal of Sport Science
13 (3), 241–248.
Sæbø, O. D., 2015. Modelling clubs financial investments in association football players. Master’s thesis at Norwegian University of Science and Technology.
Sæbø, O. D., Hvattum, L. M., 2015. Evaluating the efficiency of the association football transfer
market using regression based player ratings. Norsk Informatikkonferanse (NIK).
URL http://ojs.bibsys.no/index.php/NIK/article/view/246
Santı́n, D., 2014. Measuring the technical efficiency of football legends: who were real madrid’s
all-time most efficient players? International Transactions in Operational Research 21 (3),
439–452.
Schuckers, M., Curro, J., 2013. Total hockey rating (thor): A comprehensive statistical rating
of national hockey league forwards and defensemen based upon all on-ice events. In: 7th
Annual MIT Sloan Sports Analytics Conference.
Seçkin, A., 2006. Home advantage in association football: evidence from Turkish Super
League. In: ECOMOD Conference in Hong Kong, China, June. pp. 28–30.
Seçkin, A., Pollard, R., 2008. Home advantage in Turkish professional soccer. Perceptual and
motor skills 107 (1), 51–54.
Singh, P. K., Ahmad, M., 2015. Performance prediction of players in sports league matches.
International Journal of Science and Research (ISJR) 4 (4), 2207–2213.
Taylor, J. B., James, N., Mellalieu, S. D., 2005. Notational analysis of corner kicks in English Premier League soccer. In: Science and Football V: the Proceedings of the Fifth World
Congress on Football. pp. 229–234.
Taylor, J. B., Mellalieu, S. D., James, N., Shearer, D. A., 2008. The influence of match location,
quality of opposition, and match status on technical performance in professional association
football. Journal of Sports Sciences 26 (9), 885–895.
Tenga, A., Holme, I., Ronglan, L., Bahr, R., 2010a. Effects of match location on playing tactics
for goal scoring in norwegian professional soccer. Journal of Sport Behavior 33 (1), 89.
Tenga, A., Holme, I., Ronglan, L. T., Bahr, R., 2010b. Effect of playing tactics on achieving
score-box possessions in a random series of team possessions from norwegian professional
soccer matches. Journal of Sports Sciences 28 (3), 245–255.
85
Tenga, A., Holme, I., Ronglan, L. T., Bahr, R., 2010c. Effect of playing tactics on goal scoring
in norwegian professional soccer. Journal of Sports Sciences 28 (3), 237–244.
Tenga, A., Ronglan, L. T., Bahr, R., 2010d. Measuring the effectiveness of offensive match-play
in professional soccer. European Journal of Sport Science 10 (4), 269–277.
Thomas, A., Ventura, S. L., Jensen, S. T., Ma, S., et al., 2013. Competing process hazard
function models for player ratings in ice hockey. The Annals of Applied Statistics 7 (3),
1497–1524.
Thomas, A. C., 2006. The impact of puck possession and location on ice hockey strategy. Journal of Quantitative Analysis in Sports 2 (1), 1–16.
Transfermarkt, 2016. www.transfermarkt.com, Online; accessed 20-April-2016.
Tunaru, R., Clark, E., Viney, H., 2005. An option pricing framework for valuation of football
players. Review of financial economics 14 (3), 281–295.
Tunaru, R. S., Viney, H. P., 2010. Valuations of soccer players from statistical performance data.
Journal of Quantitative Analysis in Sports 6 (2), 1–23.
Unkelbach, C., Memmert, D., et al., 2008. Game management, context effects, and calibration:
the case of yellow cards in soccer. Journal of Sport and Exercise Psychology 30 (1), 95.
Vecer, J., 2014. Crossing in soccer has a strong negative impact on scoring: evidence from
the English Premier League, the German Bundesliga and the World Cup 2014. Available at
SSRN 2225728, http://ssrn.com/abstract=2225728.
Vecer, J., Kopriva, F., Ichiba, T., 2009. Estimating the effect of the red card in soccer: when
to commit an offense in exchange for preventing a goal opportunity. Journal of Quantitative
Analysis in Sports 5 (1), Article 8.
VG, 2014. http://vglive.no/#85585=&eliteguiden=s-borsen-sid-669,
Online; accessed 20-April-2016.
VG, 2015. http://vglive.no/#85585=&eliteguiden=s-borsen-sid-706,
Online; accessed 20-April-2016.
Wei, X., Sridharan, S., Lucey, P., Navarathna, R., Morgan, S., 2014. Analyzing and predicting
events in soccer and tennis using spatiotemporal data. In: 2014 KDD Workshop on LargeScale Sports Analytics.
URL http://eprints.qut.edu.au/78738/
Wikipedia, 2016. Logistic Regression — Wikipedia, the free encyclopedia. https://en.
wikipedia.org/wiki/Logistic_regression, Online; accessed 24-May-2016.
Wright, C., Atkins, S., Polman, R., Jones, B., Sargeson, L., 2011. Factors associated with goals
and goal scoring opportunities in professional soccer. International Journal of Performance
Analysis in Sport 11 (3), 438–449.
Wright, M., 2015. Operational Research Applied to Sports. Palgrave Macmillan, 175 Fifth Avenue, New Yotk, NY, 10010.
86
Yamada, H., Hayashi, Y., 2015. Characteristics of goal-scoring crosses in international soccer
tournaments. Football Science 12, 24–32.
87
88
Appendix
A
Tables from Tippeligaen
Appendix A serves as background information on how the 2014 and 2015 seasons of Tippeligaen ended, included for reference to the player ratings from the three developed models. Final
league tables, top scorers and players with the most assists are presented in Tables A.1 to A.6.
A.1
2014
Table A.1: League Table Tippeligaen 2014
Position Team
Games Played Won
Drawn
Lost
Goals For
Goals Against
Goal Difference
Points
1
Molde
30
22
5
3
62
24
38
71
2
Rosenborg
30
18
6
6
64
43
21
60
3
Odd
30
17
7
6
52
32
20
58
4
Strømsgodset 30
15
5
10
48
42
6
50
5
Lillestrøm
30
13
7
10
49
35
14
46
6
Vålerenga
30
11
9
10
59
53
6
42
7
Aalesund
30
11
8
11
40
39
1
41
8
Sarpsborg 08
30
10
10
10
41
48
-7
40
9
Stabæk
30
11
6
13
44
52
-8
39
10
Viking
30
8
12
10
42
42
0
36
11
Haugesund
30
10
6
14
43
49
-6
36
12
Start
30
10
5
15
47
60
-13
35
13
Bodø/Glimt
30
10
5
15
45
60
-15
35
14
Brann
30
8
5
17
41
54
-13
29
15
Sogndal
30
6
6
18
31
49
-18
24
16
Sandnes Ulf
30
4
10
16
27
53
-26
22
89
Table A.2: Top 20 players with respect to goals 2014
Position
Player
Team
Goals
Games Played
Average
1
Vidar Örn Kjartansson
Vålerenga
25
29
0.86
2
Christian Gytkjær
Haugesund
15
26
0.58
3
Alexander Søderlund
Rosenborg
13
23
0.57
4
Franck Boli
Stabæk
13
28
0.46
5
Abdurahim Laajab
Bodø/Glimt
13
30
0.43
6
Mohamed Elyounoussi
Molde
13
30
0.43
7
Frode Johnsen
Odd
11
30
0.37
8
Leke James
Aalesund
10
23
0.43
9
Fredrik Gulbrandsen
Molde
10
23
0.43
10
Péter Kovács
Strømsgodset 10
24
0.42
11
Maic Sema
Haugesund
10
26
0.38
12
Daniel Chima Chukwu
Molde
10
27
0.37
13
Fredrik Brustad
Stabæk
10
30
0.33
14
Pálmi Rafn Pálmason
Lillestrøm
9
27
0.33
15
Mahatma Otoo
Sogndal
9
27
0.33
16
Jone Samuelsen
Odd
9
28
0.32
17
Vidar Nisja
Viking
9
28
0.32
18
Jakob Orlov
Brann
9
29
0.31
19
Ghayas Zahid
Vålerenga
9
29
0.31
20
Mike Jensen
Rosenborg
9
29
0.31
90
Table A.3: Top 20 players with respect to assist 2014
Position
Player
Team
Assists
Games Played
Average
1
Bjørn Helge Riise
Lillestrøm
10
27
0.37
2
Petter Vaagan Moen
Lillestrøm
9
27
0.33
3
Morten Gamst Pedersen
Rosenborg
8
24
0.33
4
Hjörtur Logi Valgardsson
Sogndal
8
26
0.31
5
Zlatko Tripić
Start
8
28
0.29
6
Jone Samuelsen
Odd
8
28
0.29
7
Christian Grindheim
Vålerenga
8
29
0.28
8
Martin Ødegaard
Strømsgodset
7
23
0.30
9
Per-Egil Flo
Molde
7
24
0.29
10
Michael Francis Stephens
Stabæk
7
30
0.23
11
Pål André Helland
Rosenborg
6
21
0.29
12
Lars-Christopher Vilsvik
Strømsgodset
6
25
0.24
13
Håvard Storbæk
Odd
6
27
0.22
14
Mattias Moström
Molde
6
28
0.21
15
Martin Linnes
Molde
6
28
0.21
16
Petter Strand
Sogndal
6
29
0.21
17
Jón Dadi Bödvarsson
Viking
6
29
0.21
18
Papa Alioune Ndiaye
Bodø/Glimt
6
30
0.20
19
Fredrik Brustad
Stabæk
6
30
0.20
20
Magne Hoseth
Stabæk
5
9
0.56
91
A.2
2015
Table A.4: League Table Tippeligaen 2015
Position Team
Games Played Won
Drawn
Lost
Goals For
Goals Against
Goal Difference
Points
1
Rosenborg
30
21
6
3
73
27
46
69
2
Strømsgodset 30
17
6
7
67
44
23
57
3
Stabæk
30
17
5
8
54
43
11
56
4
Odd
30
15
10
5
61
41
20
55
5
Viking
30
17
2
11
53
39
14
53
6
Molde
30
15
7
8
62
31
31
52
7
Vålerenga
30
14
7
9
49
41
8
49
8
Lillestrøm
30
12
9
9
45
43
2
44
9
Bodø/Glimt
30
12
4
14
53
56
-3
40
10
Aalesund
30
11
5
14
42
57
-15
38
11
Sarpsborg 08
30
8
10
12
37
49
-12
34
12
Haugesund
30
8
7
15
33
52
-19
31
13
Tromsø
30
7
8
15
36
50
-14
29
14
Start
30
5
7
18
35
64
-29
22
15
Mjøndalen
30
4
9
17
38
69
-31
21
16
Sandefjord
30
4
4
22
36
68
-32
16
92
Table A.5: Top 20 players with respect to goals 2015
Position Player
Team
Goals
Games Played
Average
1.
Alexander Søderlund
Rosenborg
22
27
0.81
2.
Adama Diomandé
Stabæk
17
21
0.81
3.
Olivier Occéan
Odd
15
27
0.56
4.
Ola Kamara
Molde
14
29
0.48
5.
Pål André Helland
Rosenborg
13
21
0.62
6.
Alexander Sørloth
Bodø/Glimt
13
26
0.50
7.
Trond Olsen
Bodø/Glimt
13
29
0.45
8.
Leke James
Aalesund
13
29
0.45
9.
Mohamed Elyounoussi
Molde
12
28
0.43
10.
Marcus Pedersen
Strømsgodset
11
10
1.10
11.
Veton Berisha
Viking
11
14
0.79
12.
Fred Friday
Lillestrøm
11
26
0.42
13.
Iver Fossum
Strømsgodset
11
30
0.37
14.
Erling Knudtzon
Lillestrøm
10
29
0.34
15.
Ernest Asante
Stabæk
10
30
0.33
16.
Christian Gytkjær
Haugesund
10
30
0.33
17.
Tommy Høiland
Molde
9
23
0.39
18.
Fredrik Nordkvelle
Odd
9
24
0.38
19.
Zdenek Ondrásek
Tromsø
9
27
0.33
20.
Matthı́as Vilhjálmsson
Rosenborg/Start
9
27
0.33
93
Table A.6: Top 20 players with respect to assist 2015
Position
Player
Team
Assists
Games Played
Average
1.
Mike Jensen
Rosenborg
13
29
0.45
2.
Yassine El Ghanassy
Stabæk
11
26
0.42
3.
Ernest Asante
Stabæk
11
30
0.37
4.
Christian Grindheim
Vålerenga
11
30
0.37
5.
Anders Trondsen
Sarpsborg 08
9
24
0.38
6.
Espen Ruud
Odd
9
28
0.32
7.
Fredrik Midtsjø
Rosenborg
9
29
0.31
8.
Olivier Occéan
Odd
8
27
0.30
9.
Gustav Wikheim
Strømsgodset 8
28
0.29
10.
Pål André Helland
Rosenborg
7
21
0.33
11.
Daniel José Bamberg
Haugesund
7
25
0.28
12.
Per-Egil Flo
Molde
7
25
0.28
13.
Joackim Olsen Solberg
Mjøndalen
7
28
0.25
14.
Bentley
Odd
7
30
0.23
15.
Lars-Christopher Vilsvik
Strømsgodset 6
24
0.25
16.
Øyvind Storflor
Strømsgodset 6
27
0.22
17.
Matthı́as Vilhjálmsson
Rosenborg
6
27
0.22
18.
Christian Andreas Gauseth Mjøndalen
6
28
0.21
19.
Trond Olsen
Bodø/Glimt
6
29
0.21
20.
André Danielsen
Viking
6
30
0.20
94
Appendix
B
Results from Expected Goals Model
Appendix B presents additional results from running the xG model. First, the correlation matrix
for all variables in the model for regular shots is presented. Thereafter, all players scoring more
than 8 goals in Tippeligaen 2014 are presented and rated by G/xG. The results from 2015 are
repeated for comparison in Table B.4. In addition, two team tables are presented showing xG
difference and goal difference for 2014 and 2015 in Table B.3 and B.5, respectively.
Table B.1: xG Model: Correlation matrix
Length
Angle
Fast break
Take ons
Header
Cross
Through ball
Match status
Home/Away
Length
1
Angle
-0.7876
Fast break
-0.0052 -0.0169
1
Take ons
0.0575
-0.0956
0.0086
1
Header
-0.4975
0.4598
-0.0493
-0.1043
1
Cross
-0.5278
0.4501
-0.0432
-0.1142
0.6623
1
Through ball
-0.0501
-0.0167
0.0371
-0.0093
-0.0535
-0.0655
Match status
-0.0299
0.0244
0.0806
0.0367
-0.0586 -0.0393
0.0486
1
Home/Away
-0.0301 0.0290
-0.0202
-0.0147
0.0334
0.0224
-0.0066
0.1242
1
Elo diff.
-0.0492
0.0121
0.0057
0.0083
0.0157
-0.0044
0.2273
0.0193
Elo diff.
1
0.0301
1
1
95
B.1
2014
Table B.2: xG Model: All players scoring more than 8 goals in Tippeligaen 2014, ranked by G/xG
*
Name
Team
Position
Minutes played Goals xG
Shots xG/shots Goals/xG
Vidar Nisja
Viking
Attacking midfielder
1922
9
4.63
49
0.09
1.94
Jone Samuelsen
Odd
Central midfielder
2498
9
5.57
66
0.08
1.61
Fredrik Gulbrandsen
Molde
Forward
1475
10
7.00
55
0.13
1.43
Papa Alioune Ndiaye*
Bodø/Glimt
Attacking midfielder
2555
9
6.70
98
0.07
1.34
Ernest Asante
Start
Winger
1933
8
5.97
65
0.09
1.34
Tommy Høiland
Molde/Lillestrøm Forward
672
8
6.20
31
0.20
1.29
Riku Riski
Rosenborg
Winger
2174
8
6.30
61
0.10
1.27
Péter Kovács
Strømsgodset
Forward
1429
10
8.03
45
0.18
1.25
Frode Johnsen
Odd
Forward
2588
11
8.84
67
0.13
1.24
Jakob Orlov
Brann
Forward
2248
9
7.52
66
0.11
1.20
Mike Jensen
Rosenborg
Central midfielder
2590
9
7.53
81
0.09
1.19
Herolind Shala*
Odd
Midfielder
2093
9
7.64
78
0.10
1.18
Franck Boli
Stabæk
Forward
2249
13
11.48
65
0.18
1.13
Fredrik Brustad*
Stabæk
Winger
2261
10
9.18
50
0.18
1.09
Vidar Örn Kjartansson*
Vålerenga
Forward
2608
25
29.99
117
0.20
1.09
Ghayas Zahid
Vålerenga
Attacking midfielder
2216
9
8.34
57
0.15
1.08
Alexander Søderlund*
Rosenborg
Forward
1614
13
12.18
49
0.25
1.07
Christian Gytkjær
Haugesund
Forward
2001
15
14.86
71
0.21
1.01
Mohamed Elyounoussi
Molde
Winger
2533
13
13.14
107
0.12
0.99
Mahatma Otoo
Sogndal
Forward
2077
9
9.15
79
0.12
0.98
Maic Sema*
Haugesund
Attacking midfielder
1803
10
10.29
51
0.20
0.97
Diego Iván Rubio**
Sandnes Ulf
Forward
1794
8
8.32
66
0.13
0.96
Pálmi Rafn Pálmason
Lillestrøm
Attacking midfielder
2079
9
9.55
55
0.17
0.94
Abdurahim Laajab*
Bodø/Glimt
Forward
2468
13
14.43
102
0.14
0.90
Leke James
Aalesund
Forward
2031
10
11.82
76
0.16
0.85
Daniel Chima Chukwu*
Molde
Forward
1998
10
13.66
67
0.20
0.73
sold to foreign club during or some time after the 2014 season
96
**
on loan from a foreign club in 2014
Table B.3: xG Model: Tippeligaen 2014 teams rated by xG difference
*
Team
xG
xG received Real goal difference*
xG difference
Molde
67.80
19.82
38
47.99
Odd
49.21
32.38
20
16.83
Rosenborg
58.17
42.58
21
15.58
Strømsgodset
55.99
42.51
6
13.48
Lillestrøm
47.10
39.42
14
7.67
Vålerenga
54.03
48.20
6
5.82
Viking
50.44
44.96
0
5.48
Haugesund
51.83
47.42
-6
4.41
Aalesund
42.14
44.99
1
-2.85
Stabæk
43.47
52.37
-8
-8.89
Sogndal
31.87
45.83
-18
-13.97
Sandnes Ulf
35.54
50.63
-26
-15.09
Bodø/Glimt
39.98
56.37
-15
-16.40
Sarpsborg 08
33.72
50.08
-7
-17.35
Start
37.88
55.60
-13
-17.73
Brann
36.84
55.25
-13
-18.41
include own goals
97
B.2
2015
Table B.4: xG Model: All players scoring more than 8 goals in Tippeligaen 2015, ranked by G/xG
*
Name
Team
Position
Minutes played
Goals xG
Shots
xG/S G/xG
Trond Olsen
Bodø/Glimt
Winger
2510
13
6.55
74
0.09
1.98
Simon Diedhiou*
Haugesund
Forward
1945
9
4.90
52
0.09
1.84
Veton Berisha*
Viking
Forward
1256
11
6.57
47
0.14
1.67
Pål Alexander Kirkevold* Sandefjord
Forward
1881
8
5.35
66
0.08
1.50
Kristoffer Ajer*
Start
Central Midfielder
2581
8
5.43
43
0.13
1.47
Marcus Pedersen
Strømsgodset
Forward
835
11
8.20
34
0.24
1.34
Alexander Sørloth*
Bodø/Glimt
Forward
1776
13
9.89
58
0.17
1.31
Luc Kassi
Stabæk
Forward
2130
8
6.26
54
0.12
1.28
Alexander Søderlund*
Rosenborg
Forward
2242
22
17.44
87
0.20
1.26
Tobias Mikkelsen*
Rosenborg
Winger
1931
8
6.44
69
0.09
1.24
Ernest Asante
Stabæk
Winger
2643
10
8.06
70
0.12
1.24
Adama Diomandé*
Stabæk
Forward
1850
17
13.92
79
0.18
1.22
Zdeněk Ondrášek*
Tromsø
Forward
2366
9
7.57
74
0.10
1.19
Tommy Høiland
Molde
Forward
1103
9
7.93
36
0.22
1.13
Matthı́as Vilhjálmsson
Start/Rosenborg Forward
2087
9
7.99
46
0.17
1.13
Mohamed Elyounoussi
Molde
Winger
2275
12
10.69
84
0.13
1.12
Erling Knudtzon
Lillestrøm
Forward
2595
10
8.91
51
0.17
1.12
Fredrik Nordkvelle
Odd
Attacking Midfielder
1891
9
8.06
48
0.17
1.12
Iver Fossum*
Strømsgodset
Attacking Midfielder
2653
11
10.07
69
0.15
1.09
Pål André Helland
Rosenborg
Winger
1502
13
12.07
89
0.14
1.08
Suleiman Abdullahi
Viking
Forward
1882
8
7.77
70
0.11
1.03
Christian Gytkjær
Haugesund
Forward
2545
10
9.84
57
0.17
1.02
Sander Svendsen
Molde
Forward
1720
8
8.13
60
0.14
0.98
Fred Friday
Lillestrøm
Forward
1770
11
11.22
67
0.17
0.98
Ola Kamara*
Molde
Forward
2384
14
14.69
89
0.17
0.95
Bentley
Odd
Winger
2464
8
8.53
67
0.13
0.94
Leke James*
Aalesund
Forward
2610
13
13.97
91
0.15
0.93
Jón Dadi Bödvarsson*
Viking
Forward
2067
9
10.06
58
0.17
0.89
Gustav Wikheim*
Strømsgodset
Winger
2413
9
11.35
68
0.17
0.79
Olivier Occéan
Odd
Forward
2208
15
20.23
89
0.23
0.74
sold to foreign club during or after the 2015 season
98
**
on loan from a foreign club in 2015
Table B.5: xG Model: Tippeligaen 2015 teams rated by xG difference
*
Team
xG
xG received Real goal difference* xG difference
Rosenborg
68.66
24.20
46
43.46
Molde
60.32
30.09
31
30.23
Strømsgodset
62.69
33.55
23
29.13
Odd
61.92
34.37
20
27.55
Stabæk
53.67
36.09
11
17.57
Viking
54.56
39.53
14
15.03
Vålerenga
47.56
42.20
8
5.36
Lillestrøm
51.37
49.39
2
1.98
Sarpsborg 08
34.60
41.65
-12
-7.05
Bodø/Glimt
38.86
48.24
-3
-9.38
Tromsø
31.28
46.40
-14
-15.12
Haugesund
30.92
51.44
-19
-20.52
Mjøndalen
38.83
60.22
-31
-21.38
Sandefjord
32.40
59.03
-32
-26.62
Start
32.02
62.42
-29
-30.41
Aalesund
36.43
69.75
-15
-33.32
include own goals
99
100
Appendix
C
Results from Markov Game Model 1
Appendix C presents additional results from running Model 1. Top 10 lists for the different
playing positions for Tippeligaen 2014 are presented in Tables C.1 and C.2. In Tables C.4 and
C.5 are the results for 2015 repeated for comparison. Tables C.3 and C.6 show the league tables
that can be obtained for 2014 and 2015, respectively.
101
C.1
2014
Table C.1: Model 1: Top 10 players in offensive positions, Tippeligaen 2014.
Average value = 0.1118, minimum value = -0.2324
FORWARDS
Player
Team
Minutes
Total
Shot
Take on
Foul won
Corner won
Aerial
Ball carry
Vidar Örn Kjartansson*
Vålerenga
2608
0.4184
Fredrik Gulbrandsen
Molde
1475
0.3399
0.3027
0.0020
0.0222
0.0050
-0.0044
0.0696
0.1846
0.0528
0.0156
0.0054
0.0004
0.0807
Alexander Søderlund*
Rosenborg
1614
Franck Boli
Stabæk
2249
0.3332
0.2981
0.0066
0.0060
0.0018
-0.0009
0.0180
0.3190
0.1366
0.0509
0.0134
0.0447
0.0037
0.0478
Christian Gytkjær
Haugesund
2001
0.2727
Abdurahim Laajab*
Bodø/Glimt
2468
0.1979
0.2231
-0.0055
0.0066
0.0056
0.0157
0.0181
0.0782
0.0022
0.0177
0.0414
0.0113
0.0426
Daniel Chima Chukwu*
Molde
1998
0.1937
0.0058
0.0271
0.0169
0.0058
0.0022
0.0389
Péter Kovács
Strømsgodset
1429
0.1887
0.1849
-
0.0030
0.0012
0.0200
0.0308
Frode Johnsen
Odd
2588
0.1661
0.0490
-0.0009
0.0123
0.0043
0.0261
0.0100
Diego Iván Rubio**
Sandnes Ulf
1794
0.1604
0.0575
-0.0020
0.0212
0.0061
-0.0106
0.0532
0.2590
0.1520
0.0133
0.0135
0.0121
0.0064
0.0410
Top 10 average
WINGERS
Player
Team
Minutes
Total
Shot
Pass
Take on
Cross
Corner
Carry
Pål André Helland
Rosenborg
1040
0.3358
0.0298
0.1083
0.0350
0.0878
0.0084
0.0642
Ernest Asante
Start
1933
0.3206
0.1039
0.0731
0.0210
0.0203
-
0.0451
Gustav Wikheim
Strømsgodset 1616
0.2966
0.1089
0.0792
0.0109
0.0506
-0.0017
-0.0009
Zlatko Tripić*
Start
2107
0.2639
0.0144
0.0035
-0.0064
0.0109
0.0484
0.0482
Fredrik Brustad*
Stabæk
2261
0.2471
0.1148
0.0224
0.0212
0.0158
-
0.0520
Øyvind Storflor
Strømsgodset 2029
0.2432
0.0282
0.0841
-0.0039
0.0437
0.0181
0.0211
Riku Riski
Rosenborg
2174
0.2177
0.0276
0.0191
0.0147
0.0271
-0.0005
0.0779
Dane Richards*
Bodø/Glimt
1181
0.2175
0.0615
-0.0004
0.0140
0.0645
0.0266
0.0029
Mohamed Elyounoussi
Molde
2533
0.2132
0.0164
0.0263
0.0043
0.0076
-0.0060
0.0918
Mattias Moström
Molde
2152
0.1875
-0.0387
0.0630
0.0117
0.0109
0.0755
0.0083
0.2543
0.0467
0.0479
0.0123
0.0339
0.0169
0.0411
Top 10 average
ATTACKING MIDFIELDERS
Player
Team
Total
Shot
Pass
Take on
Foul won
Cross
Carry
Martin Ødegaard*
Strømsgodset 1454
Minutes
0.4030
0.1472
0.1340
0.0038
0.0130
0.0112
0.0154
Maic Sema*
Haugesund
1803
0.2842
0.1967
0.0071
0.0143
0.0025
0.0037
0.0231
Ghayas Zahid
Vålerenga
2216
0.2410
0.0830
0.0625
0.0078
0.0131
-0.0085
0.0485
Petter Vaagan Moen
Lillestrøm
1915
0.2327
0.1107
0.0189
-0.0026
0.0129
0.0164
0.0108
Vidar Nisja
Viking
1922
0.2263
0.1285
0.0073
0.0015
0.0079
-0.0162
0.0601
Herolind Shala*
Odd
2093
0.2163
0.0615
0.0337
0.0062
0.0178
0.0633
0.0259
Håvard Storbæk
Odd
1497
0.2022
0.0618
0.0426
0.0097
0.0086
0.0583
0.0041
Papa Alioune Ndiaye*
Bodø/Glimt
2555
0.1883
0.0039
0.0389
0.0566
0.0092
-0.0036
0.0410
Pálmi Rafn Pálmason
Lillestrøm
2079
0.1801
0.0738
-0.0210
0.0058
0.0565
-0.0034
0.0307
Michael Barrantes*
Aalesund
1608
0.1781
-0.0034
0.0869
0.0269
0.0064
-0.0059
0.0069
0.2352
0.0864
0.0411
0.0130
0.0148
0.0115
0.0267
Top 10 average
*
sold to foreign club during or some time after the 2014 season
102
**
on loan from a foreign club in 2014
Table C.2: Model 1: Top 10 players in defensive positions, Tippeligaen 2014.
Average value = 0.1118, minimum value = -0.2324
CENTRAL MIDFIELDERS
Player
Team
Minutes
Total
Shot
Pass
Tackle
Ball rec.
Cross
Corner
Jone Samuelsen
Odd
2498
0.3338
0.0948
0.0741
0.0136
0.0098
0.0857
-0.0017
Sakari Mattila*
Aalesund
1688
0.3123
0.2082
0.0178
0.0141
0.0086
0.0011
0.0282
Fredrik Midtsjø
Rosenborg
1754
0.2535
0.0700
0.0379
0.0099
0.0059
0.0292
0.0024
Christian Grindheim
Vålerenga
2519
0.2429
0.0598
0.1136
0.0147
0.0102
0.0065
0.0199
Mike Jensen
Rosenborg
2590
0.2299
0.0636
0.0466
0.0102
0.0087
0.0261
0.0046
Makhtar Thioune*
Viking
1115
0.2198
0.0326
0.1161
0.0159
0.0041
0.0002
0.0405
Björn Danı́el Sverrisson
Viking
2275
0.2036
0.0618
0.0810
0.0090
0.0059
0.0050
-0.0024
Daniel Berg Hestad
Molde
1255
0.1549
-0.0178
0.1549
0.0108
0.0097
0.0175
-0.0005
Peter Orry Larsen
Aalesund
1887
0.1402
0.0741
0.0112
0.0031
0.0030
0.0049
-0.0008
Ole Kristian Selnæs
Rosenborg
1770
0.1310
0.0120
0.0359
0.0207
0.0127
0.0146
0.0556
0.2222
0.0659
0.0689
0.0122
0.0079
0.0191
0.0146
Top 10 average
FULL BACKS
Player
Team
Minutes
Total
Shot
Pass
Tackle
Clearance
Ball rec.
Cross
Per-Egil Flo
Molde
1959
0.2597
-0.0166
0.0774
0.0083
0.0127
0.0102
0.0629
Jarkko Hurme
Odd
1018
0.2543
0.0458
0.0093
0.0093
0.0295
0.0063
0.1244
Claes Phillip Kronberg
Sarpsborg 08 2501
0.2242
0.0861
0.0484
0.0159
0.0147
0.0142
0.0196
Mikael Dorsin
Rosenborg
2072
0.2154
0.0662
0.0616
0.0131
0.0133
0.0091
0.0177
Martin Linnes*
Molde
2520
0.1991
0.0221
0.0552
0.0130
0.0166
0.0167
0.0629
Andreas Vindheim
Brann
1874
0.1952
0.0038
0.0027
0.0279
0.0095
0.0108
0.0571
Amin Nouri
Start
1482
0.1898
0.0388
0.0404
0.0214
0.0100
0.0132
0.0406
Kristoffer Haugen
Viking
1717
0.1752
0.0273
0.0384
0.0244
0.0085
0.0146
0.0292
Jonas Svensson
Rosenborg
1934
0.1728
0.0419
0.0765
0.0228
0.0098
0.0097
0.0009
André Danielsen
Viking
2474
0.1643
-0.0129
0.0392
0.0179
0.0190
0.0134
0.0471
0.2050
0.0303
0.0449
0.0174
0.0144
0.0118
0.0462
Top 10 average
CENTRE BACKS
Player
Team
Minutes
Total
Shot
Pass
Foul won
Tackle
Clearance
Ball rec.
Vegard Forren
Molde
2467
0.2217
0.0344
0.0902
0.0055
0.0118
0.0425
0.0147
Jonas Grønner
Brann
1711
0.1853
0.0501
0.0486
0.0067
0.0042
0.0089
-0.0005
Brede Moe
Bodø/Glimt
2250
0.1636
0.0559
0.0543
0.0052
0.0113
0.0296
0.0020
Jon Inge Høiland
Stabæk
1873
0.1613
0.0967
0.0528
0.0026
0.0048
0.0044
0.0104
Indridi Sigurdsson
Viking
2326
0.1565
0.0491
0.0537
0.0018
0.0067
0.0134
0.0109
Martin Jensen
Sarpsborg 08
1874
0.1489
0.0514
0.0672
0.0083
0.0101
0.0143
0.0076
Jonatan Tollås Nation
Aalesund
1036
0.1383
0.0482
0.0649
0.0078
0.0135
0.0299
0.0052
Simon Andreas Larsen
Vålerenga
2691
0.1352
0.0503
0.0590
0.0025
0.0051
0.0071
0.0079
Nikolaj Høgh
Vålerenga
1192
0.1324
0.0067
0.0658
0.0033
0.0155
0.0166
0.0076
Marius Amundsen
Lillestrøm
1216
0.1323
0.0521
0.0484
0.0034
0.0062
0.01986
0.0036
0.1575
0.0495
0.0605
0.0047
0.0089
0.0186
0.0069
Top 10 average
*
sold to foreign club during or some time after the 2014 season
**
on loan from a foreign club in 2014
103
Table C.3: Model 1: League table 2014 based on the model compared to the real table
104
Team
Value/match
Table
Real table
Diff
Molde
1.1825
1
1
0
Rosenborg
0.5863
2
2
0
Odd
0.4795
3
3
0
Lillestrøm
0.3836
4
5
-1
Vålerenga
0.2055
5
6
-1
Strømsgodset
0.0502
6
4
2
Stabæk
0.0367
7
9
-2
Aalesund
0.0340
8
7
1
Viking
-0.0016
9
10
-1
Sarpsborg 08
-0.1370
10
8
2
Haugesund
-0.1618
11
11
0
Start
-0.3338
12
12
0
Brann
-0.4886
13
14
-1
Bodø/Glimt
-0.5749
14
13
1
Sandnes Ulf
-0.6263
15
16
-1
Sogndal
-0.6343
16
15
1
C.2
2015
Table C.4: Model 1: Top 10 players in offensive positions, Tippeligaen 2015.
Average value = 0.1124, minimum value = -0.1248
FORWARDS
Player
Team
Minutes
Total
Shot
Pass
Take on
Foul won
Cross
Carry
Adama Diomandé*
Stabæk
1850
0.5448
0.2736
0.0252
0.0287
0.0317
0.0454
0.0519
Veton Berisha*
Viking
1256
0.4721
0.3702
0.0475
-0.0167
0.0063
0.0097
0.0562
Alexander Søderlund*
Rosenborg
2242
0.3896
0.3674
-0.0244
0.0003
0.0094
0.0047
0.0154
Tommy Høiland
Molde
1103
0.3771
0.2945
0.0077
0.0175
0.0045
0.0090
0.0099
Fred Friday
Lillestrøm
1770
0.3130
0.1145
-0.0011
0.0574
0.0207
0.0034
0.0954
Alexander Sørloth*
Bodø/Glimt
1776
0.3108
0.2304
-0.0481
0.0258
0.0154
0.0091
0.0250
Luc Kassi
Stabæk
2130
0.2759
0.1064
0.1279
0.0156
0.0140
0.0051
-0.0036
Erling Knudtzon
Lilestrøm
2595
0.2481
0.1051
0.0665
0.0139
0.0118
0.0030
0.0374
Simon Diédhiou*
Haugesund
1945
0.2327
0.1287
0.0097
0.0285
0.0129
-0.0224
0.0431
Matthı́as Vilhjálmsson
Start/Rosenborg
2087
0.1961
0.0941
0.0450
0.0047
0.0060
0.0146
0.0317
0.3360
0.2085
0.0256
0.0175
0.0133
0.0082
0.0362
Top 10 average
WINGERS
Player
Team
Minutes
Total
Shot
Pass
Take on
Foul won
Cross
Carry
Pål André Helland
Rosenborg
1502
0.5553
0.2431
0.0381
0.0055
0.0357
0.0021
0.1153
Trond Olsen
Bodø/Glimt
2510
0.3749
0.1932
0.0374
0.0244
0.0110
0.0503
0.0503
Moryké Fofana*
Lillestrøm
1300
0.3683
0.1777
0.0728
0.0400
0.0109
-0.0013
0.0098
Zymer Bytyqi
Viking
1258
0.3111
0.0527
0.0236
0.0130
0.0043
0.1469
0.0315
Ernest Asante
Stabæk
2643
0.2961
0.0470
0.0850
0.0226
0.0172
0.0206
0.0741
Gustav Wikheim*
Strømsgodset
2413
0.2935
-0.0084
0.1016
0.0708
0.0037
0.0528
0.0439
Espen Børufsen
Start
2154
0.2209
0.0815
-0.0075
0.0107
0.0084
0.0619
0.0145
Ole Jørgen Halvorsen
Odd
1393
0.2160
0.0144
0.0047
-0.0057
0.0038
0.1064
0.0295
Magnus Andersen
Tromsø
2663
0.1950
0.0428
0.0321
-0.0030
0.0036
0.0343
0.0374
Mohamed Elyounoussi
Molde
2275
0.1855
0.0892
0.0513
-0.0051
0.0185
-0.0146
0.0165
0.3017
0.0933
0.0439
0.0173
0.0117
0.0459
0.0423
Top 10 average
ATTACKING MIDFIELDERS
Player
Team
Minutes
Total
Shot
Pass
Cross
Corner
Carry
FK
Fredrik Nordkvelle
Odd
1891
0.2719
0.1954
0.0041
0.0174
-0.0011
-0.0069
-0.0006
Daniel Fredheim Holm
Vålerenga
1974
0.2687
0.1749
0.0404
0.0097
0.0026
-0.0116
0.0021
Michael Barrantes*
Aalesund
1000
0.2261
0.0503
0.0358
-0.0020
0.0122
0.0397
0.0297
Eirik Hestad
Molde
941
0.2161
0.0232
0.0963
0.0041
0.0195
-0.0072
0.0381
Papa Alioune Ndiaye*
Bodø/Glimt
1286
0.2087
-0.0279
0.0383
0.0189
0.0098
0.0321
-0.0010
Ghayas Zahid
Vålerenga
2383
0.2055
0.0082
0.0851
-0.0040
-0.0007
0.0479
-
Iver Fossum*
Strømsgodset
2653
0.2045
0.0744
0.0371
0.0174
-0.0003
0.0226
-0.0030
Gjermund Åsen
Tromsø
1709
0.2026
-0.0156
0.0095
0.0334
0.0814
0.0355
0.0277
Henrik Furebotn
Bodø/Glimt
2157
0.1731
0.0769
-0.0008
0.0395
0.0030
0.0047
0.0241
Thomas Kind Bendiksen*
Molde
931
0.1407
-0.0503
0.0233
0.0156
0.0900
0.0084
0.0332
0.2118
0.0509
0.0369
0.0150
0.0216
0.0165
0.0150
Top 10 average
*
sold to foreign club during or some time after the 2015 season
**
on loan from a foreign club in 2015
105
Table C.5: Model 1: Top 10 players in defensive positions, Tippeligaen 2015.
Average value = 0.1124, minimum value = -0.1248
CENTRAL MIDFIELDERS
Player
Team
Minutes
Total
Shot
Christian Grindheim
Vålerenga
2675
0.3087
Malaury Martin
Lillestrøm
975
0.2947
Giorgi Gorozia
Stabæk
1467
0.2326
-0.0578
0.1369
0.0065
0.0241
0.0063
0.0858
Kristoffer Ajer*
Start
2581
0.2187
0.1016
0.0243
0.0077
0.0122
0.0066
-
Bismark Adjei-Boateng**
Strømsgodset 1293
0.1847
0.0590
0.0192
0.0214
0.0026
0.0440
-0.0002
Morten Konradsen
Bodø/Glimt
0.1740
0.1439
-0.0058
0.0044
0.0071
0.0202
-0.0009
1362
Pass
Tackle
Ball rec
Cross
Corner
0.0985
0.1239
0.0069
0.0105
0.0040
0.0125
0.1446
0.0453
0.0045
0.0088
0.0103
0.0336
Kamal Issah
Stabæk
1846
0.1733
0.0274
0.0838
0.0143
0.0163
0.0082
-
Fredrik Midtsjø
Rosenborg
2396
0.1718
0.0227
0.0833
0.0152
0.0118
0.0010
-
Johan Andersson
Lillestrøm
973
0.1664
0.1054
0.0429
0.0046
0.0117
-0.0056
-0.0014
Ole Kristian Selnæs*
Rosenborg
1951
Top 10 average
0.1553
0.0157
0.0782
0.0133
0.0217
0.0226
-0.0025
0.2080
0.0661
0.0632
0.0099
0.0127
0.0118
0.0127
FULL BACKS
Player
Team
Minutes
Total
Shot
Pass
Cross
Ball rec
Corner
FK
Per-Egil Flo
Molde
1998
0.3290
0.0396
0.1003
0.0327
0.0092
0.0700
0.0087
Espen Ruud
Odd
2476
0.2953
0.0457
0.0925
0.0831
0.0070
0.0087
0.0204
Lars-Christopher Vilsvik
Strømsgodset 2126
0.2692
0.0542
0.0197
0.1462
0.0134
0.0153
-0.0021
Jo Nymo Matland*
Aalesund
1425
0.2429
0.0605
-0.0015
0.0426
0.0049
0.0580
0.0501
André Danielsen
Viking
2700
0.2329
0.0995
0.0428
0.0459
0.0155
0.0179
0.0050
Kent-Are Antonsen
Tromsø
2248
0.2275
0.0557
0.0029
0.0277
0.0160
0.0102
0.0261
Joachim Olsen Solberg
Mjøndalen
2486
0.2087
-0.0588
0.0230
0.0192
0.0143
0.0508
0.1031
Zarek Chase Valentin*
Bodø/Glimt
2074
0.2006
0.0056
0.0112
0.0498
0.0114
-0.0025
0.0014
Jørgen Skjelvik
Rosenborg
1812
0.1855
0.0154
0.0801
0.0449
0.0120
0.0070
0.0006
Mikael Dorsin
Rosenborg
1582
0.1492
0.0365
0.0294
0.0210
0.0106
-0.0011
-0.0011
0.2341
0.0354
0.0400
0.0513
0.0114
0.0234
0.0212
Pass
Clearance
Ball rec
Cross
Long pass
Top 10 average
CENTRE BACKS
Player
Team
Minutes
Shot
Johan Bjørdal
Rosenborg
1093
0.2398
0.0461
0.1655
-0.0001
0.0107
-0.0026
0.0012
Rhett Bernstein*
Mjøndalen
1234
0.1927
0.1214
0.0301
-0.0068
0.0077
0.0002
-0.0038
Joona Toivio
Molde
1706
0.1781
-0.0190
0.0665
0.0274
0.0093
0.0650
0.0125
Andreas Nordvik*
Sarpsborg 08
1846
0.1667
0.0532
0.0586
0.0168
0.0065
0.0036
0.0403
Brede Moe
Bodø/Glimt
2408
0.1657
0.0544
0.0545
0.0304
0.0072
-0.0016
-0.0082
Lars-Kristian Eriksen
Odd
2340
0.1456
0.0235
0.0746
0.0132
0.0084
0.0003
0.0110
Ole Christoffer Heieren Hansen Sarpsborg 08
2059
0.1383
0.0634
0.0578
0.0337
0.0051
0.0000
-0.0025
Vegard Forren
Molde
2416
0.1196
-0.0375
0.0902
0.0199
0.0069
0.0052
0.0256
Morten Sundli
Mjøndalen
2018
0.1050
0.0314
0.0381
-0.0137
0.0092
0.0029
-0.0034
Steffen Hagen
Odd
2610
0.1000
0.0174
0.0813
-0.0033
0.0089
-0.0002
0.0079
0.1551
0.0354
0.0717
0.0117
0.0080
0.0073
0.0081
Top 10 average
*
Total
sold to foreign club during or some time after the 2015 season
106
**
on loan from a foreign club in 2015
Table C.6: Model 1: League table 2015 based on the model compared to the real table
Team
Value/match
Table
Real table
Diff
Rosenborg
1.3148
1
1
0
Molde
0.8748
2
6
-4
Strømsgodset
0.7249
3
2
1
Stabæk
0.4698
4
3
1
Odd
0.4641
5
4
1
Viking
0.4397
6
5
1
Vålerenga
0.3452
7
7
0
Lillestrøm
0.1848
8
8
0
Bodø/Glimt
-0.0747
9
9
0
Tromsø
-0.3564
10
13
-3
Sarpsborg 08
-0.4349
11
11
0
Aalesund
-0.4681
12
10
2
Haugesund
-0.7172
13
12
1
Start
-0.8898
14
14
0
Mjøndalen
-0.9365
15
15
0
Sandefjord
-0.9400
16
16
0
107
108
Appendix
D
Results from Markov Game Model 2
Appendix D presents additional results from running Model 2. Top 10 lists for the different
playing positions for Tippeligaen 2014 are presented in Tables D.1 and D.2. In Tables D.4 and
D.5 are the results for 2015 repeated for comparison. Tables D.3 and D.6 show the league tables
that can be obtained for 2014 and 2015, respectively.
109
D.1
2014
Table D.1: Model 2: Top 10 players in offensive positions, Tippeligaen 2014.
Average value = 0.4668, minimum value = 0.0118
FORWARDS
Player
Team
Minutes
Total
Shot
Pass
Take on
Foul won
Cross
Ball carry
Aaron Samuel*
Sarpsborg 08
1032
0.9668
0.4466
0.1850
0.0576
0.0528
0.0098
0.1126
Jón Dadi Bödvarsson*
Viking
2129
0.9221
0.4332
0.1714
0.0322
0.0165
0.0280
0.1341
Daniel Chima Chukwu*
Molde
1998
0.9213
0.4031
0.2486
0.0489
0.0204
0.0102
0.1069
Leke James*
Aalesund
2031
0.9086
0.4005
0.2146
0.0436
0.0282
0.0058
0.1326
Vidar Örn Kjartansson
Vålerenga
2608
0.8462
0.5047
0.1383
0.0152
0.0404
0.0070
0.0915
Fredrik Gulbrandsen
Molde
1475
0.8138
0.3853
0.1360
0.0572
0.0221
0.0187
0.1460
Diego Ivan Rubio**
Sandnes Ulf
1794
0.7807
0.3213
0.1904
0.0080
0.0314
0.0460
0.1077
Matthı́as Vilhjálmsson
Start
1681
0.7746
0.3531
0.2165
0.0056
0.0191
0.0163
0.0725
Erik André Huseklepp
Brann
1989
0.7592
0.1992
0.1729
0.0351
0.0158
0.1086
0.0994
Franck Boli
Stabæk
2249
0.7547
0.3451
0.1687
0.0680
0.0225
0.0027
0.1029
0.8448
0.3792
0.1842
0.0371
0.0269
0.0253
0.1106
Top 10 average
WINGERS
Player
Team
Minutes
Total
Shot
Pass
Take on
Cross
Corner
Ball carry
Trond Olsen
Bodø/Glimt
1443
1.1593
0.3857
0.1951
0.0563
0.1644
0.0852
0.1448
Mohamed Elyounoussi
Molde
2533
1.0804
0.4095
0.3019
0.0435
0.0320
0.0252
0.1798
Pål André Helland
Rosenborg
1040
1.0453
0.3037
0.2246
0.0758
0.1571
0.0307
0.1401
Elbasan Rashani*
Odd
1229
0.9340
0.3730
0.1500
0.0597
0.1047
0.0563
0.1098
Zlatko Tripić*
Start
2107
0.9238
0.2287
0.1725
0.0258
0.0777
0.0980
0.1182
Moryké Fofana*
Lillestrøm
2051
0.9187
0.2673
0.2737
0.0826
0.0335
0.0014
0.1785
Mattias Moström
Molde
2152
0.8384
0.1138
0.3682
0.0349
0.0350
0.0578
0.1489
Øyvind Storflor
Strømsgodset
2029
0.8279
0.1402
0.3235
0.0107
0.0919
0.0668
0.0854
Gustav Wikheim*
Strømsgodset
1616
0.8206
0.1549
0.3301
0.0636
0.0844
0.0005
0.1218
Morten Gamst Pedersen
Rosenborg
1724
0.7943
0.1197
0.2108
0.0147
0.0656
0.1647
0.0388
0.9343
0.2496
0.2551
0.0468
0.0846
0.0587
0.1266
Top 10 average
ATTACKING MIDFIELDERS
Player
Team
Martin Ødegaard*
Michael Barrantes*
Minutes
Total
Shot
Pass
Take on
Cross
Corner
Ball carry
Strømsgodset 1454
1.1504
0.1496
0.5469
0.0594
0.0117
0.0552
0.2207
Aalesund
1608
0.9481
0.3274
0.2714
0.0199
0.0509
0.0366
0.1092
Ghayas Zahid
Vålerenga
2216
0.8955
0.2570
0.3054
0.0347
0.0521
0.0111
0.1518
Herolind Shala*
Odd
2093
0.8760
0.3077
0.2135
0.0172
0.1131
0.0298
0.0967
Papa Alioune Ndiaye*
Bodø/Glimt
2555
0.7923
0.2930
0.1647
0.0720
0.0377
0.0496
0.0817
Bojan Zajić
Sarpsborg 08
1826
0.7890
0.2704
0.2190
0.0265
0.0261
0.0572
0.1052
Petter Vaagan Moen
Lillestrøm
1915
0.7322
0.1986
0.2174
0.0062
0.0451
0.1187
0.0534
Maic Sema*
Haugesund
1803
0.7183
0.2742
0.2175
0.0195
0.0412
0.0012
0.1120
Fredrik Nordkvelle
Odd
1110
0.6898
0.2094
0.2382
0.0312
0.0546
0.0012
0.0984
Vidar Nisja
Viking
1922
0.6572
0.2714
0.1900
0.0178
0.0222
0.0004
0.0914
0.8249
0.2559
0.2584
0.0304
0.0455
0.0361
0.1121
Top 10 average
*
sold to foreign club during or some time after the 2014 season
110
**
on loan from a foreign club in 2014
Table D.2: Model 2: Top 10 players in defensive positions, Tippeligaen 2014.
Average value = 0.4668, minimum value = 0.0118
CENTRAL MIDFIELDERS
Player
Team
Minutes
Total
Shot
Pass
Cross
Long
Corner
Ball carry
Harmeet Singh
Molde
2424
0.8135
0.1562
0.3984
0.0183
0.0699
-
0.1238
Jone Samuelsen
Odd
2498
0.7631
0.2123
0.2439
0.0718
0.0182
0.0009
0.1285
Etzaz Hussain*
Molde
1279
0.7284
0.1048
0.3832
0.0073
0.0582
0.0286
0.0939
Mike Jensen
Rosenborg
2590
0.6937
0.2325
0.2375
0.0340
0.0280
0.0088
0.0737
Fredrik Midtsjø
Rosenborg
1754
0.6386
0.1738
0.1936
0.0341
0.0146
0.0162
0.0932
Daniel Berg Hestad
Molde
1255
0.6357
0.0838
0.3833
0.0244
0.0266
0.0013
0.0990
Gudmundur Thórarinsson*
Sarpsborg 08
2392
0.6334
0.1153
0.2450
0.0265
0.0348
0.0849
0.0837
Makhtar Thioune*
Viking
1115
0.6315
0.0498
0.2933
0.0190
0.0568
0.0661
0.0747
Michael Francis Stephens*
Stabæk
2551
0.6275
0.0895
0.3010
0.0377
0.0475
0.0136
0.0968
Mix Diskerud*
Rosenborg
1762
0.6187
0.1420
0.2917
0.0086
0.0271
0.0077
0.0631
0.6784
0.1360
0.2971
0.0282
0.0382
0.0228
0.0930
Top 10 average
FULL BACKS
Player
Team
Lars-Christopher Vilsvik
Per-Egil Flo
Martin Linnes*
Minutes
Total
Shot
Pass
Cross
Corner
Throw
Ball carry
Strømsgodset 2152
0.6473
0.1104
0.2500
0.0835
0.0557
0.0356
0.0654
Molde
1959
0.6468
0.0583
0.2645
0.0753
0.1014
0.0385
0.0714
Molde
2520
0.6171
0.1086
0.2250
0.1004
-
0.0361
0.0734
André Danielsen
Viking
2474
0.4822
0.0479
0.1403
0.0899
0.0524
0.0276
0.0643
Joakim Våge Nilsen
Haugesund
2625
0.4590
0.0501
0.2143
0.0402
0.0016
0.0235
0.0553
Thomas Grøgaard
Odd
2700
0.4452
0.0131
0.2188
0.0849
0.0228
0.0449
0.0430
Andreas Vindheim*
Brann
1874
0.4445
0.0844
0.1032
0.1441
0.0069
0.0322
0.0667
Birkir Már Sævarsson
Brann
1231
0.4408
0.0991
0.1564
0.0714
-
0.0297
0.0782
Kristoffer Haugen
Viking
1717
0.4406
0.0237
0.1009
0.1267
0.0828
0.0232
0.0364
Jo Nymo Matland*
Aalesund
1067
0.4390
0.0710
0.1101
0.0892
0.0528
0.0295
0.0349
0.5062
0.0667
0.1784
0.0905
0.0376
0.0321
0.0589
Top 10 average
CENTRE BACKS
Player
Team
Minutes
Total
Shot
Pass
Aerial
Cross
Long
Ball carry
Ruben Gabrielsen
Molde
1954
0.3575
0.0989
0.1387
0.0409
0.0299
0.0264
0.0273
Vegard Forren
Molde
2467
0.2867
0.0706
0.1422
0.0142
0.0058
0.0496
0.0456
Jarl André Storbæk
Strømsgodset 1783
0.2495
0.0395
0.1260
-0.0002
0.0339
0.0235
0.0439
Lars-Kristian Eriksen
Odd
2230
0.2476
0.0411
0.1049
-0.001
0.0353
0.0233
0.0349
Azar Karadaş
Brann
1640
0.2437
0.0873
0.1134
0.0210
0.0045
0.0174
0.0224
Tor Arne Andreassen
Haugesund
2462
0.2384
0.0911
0.0975
-0.0011
0.0215
0.0189
0.0302
Stefan Strandberg*
Rosenborg
1643
0.2343
0.0957
0.1088
0.0122
0.0036
0.0406
0.0291
Marius Christopher Høibråten
Strømsgodset
1372
0.2279
0.0631
0.1386
-0.0026
0.0019
0.0244
0.0328
Daniel Mojsov*
Brann
1393
0.2092
0.1065
0.0882
0.006
0.0064
0.0295
0.0220
Tore Reginiussen
Rosenborg
2065
0.2024
0.1201
0.0904
-0.0044
0.0009
0.0164
0.0286
0.2491
0.0814
0.1149
0.0085
0.0144
0.0270
0.0317
Top 10 average
*
sold to foreign club during or some time after the 2014 season
**
on loan from a foreign club in 2014
111
Table D.3: Model 2: League table 2014 based on the model compared to the real table
112
Team
Value/match
Table
Real Table
Diff
Molde
3.6921
1
1
0
Strømsgodset
2.0150
2
4
-2
Odd
1.7909
3
3
0
Rosenborg
1.4025
4
2
2
Viking
0.5882
5
10
-5
Vålerenga
0.5035
6
6
0
Lillestrøm
0.3532
7
5
2
Brann
-0.3729
8
14
-6
Haugesund
-0.5862
9
11
-2
Sogndal
-0.7550
10
15
-5
Aalesund
-0.8503
11
7
4
Sarpsborg 08
-0.9429
12
8
4
Start
-1.3133
13
12
1
Stabæk
-1.3528
14
9
5
Bodø/Glimt
-1.7765
15
13
2
Sandnes Ulf
-2.3953
16
16
0
D.2
2015
Table D.4: Model 2: Top 10 players in offensive positions, Tippeligaen 2015.
Average value = 0.4608, minimum value = 0.0059
FORWARDS
Player
Team
Minutes
Total
Shot
Pass
Take on
Foul won
Aerial
Ball carry
Adama Diomandé*
Stabæk
1850
1.0543
0.5001
0.2130
0.0656
0.0333
0.0312
0.1160
Olivier Occéan
Odd
2208
0.9867
0.5049
0.2432
0.0128
0.0407
0.0888
0.0616
Fred Friday
Lillestrøm
1770
0.9133
0.4026
0.1939
0.0740
0.0303
0.0115
0.1419
Ola Kamara*
Molde
2384
0.9085
0.4298
0.2208
0.0143
0.0246
0.0210
0.1069
Veton Berisha*
Viking
1256
0.8831
0.3770
0.2317
0.0165
0.0232
0.0258
0.1247
Sander Svendsen
Molde
1720
0.8309
0.3897
0.1975
0.0471
0.0101
0.0047
0.1174
Jón Dadi Bödvarsson*
Viking
2067
0.8080
0.3123
0.1932
0.0326
0.0273
0.0365
0.1328
Leke James*
Aalesund
2610
0.8047
0.3676
0.1920
0.0487
0.0212
0.0609
0.0723
Alexander Søderlund*
Rosenborg
2242
0.7993
0.4564
0.1677
0.0131
0.0279
0.0491
0.0550
Alexander Sørloth*
Bodø/Glimt
1776
0.7916
0.3829
0.1769
0.0233
0.0265
0.0719
0.0762
0.8781
0.4123
0.2030
0.0348
0.0265
0.0401
0.1005
Top 10 average
WINGERS
Player
Team
Minutes
Total
Shot
Pass
Take on
Corner won
Cross
Ball carry
Pål André Helland
Rosenborg
1502
1.4047
0.4942
0.2508
0.1021
0.0379
0.0729
0.1821
Gustav Wikheim*
Strømsgodset
2413
1.1020
0.3125
0.3692
0.1477
0.0246
0.0381
0.1562
Yassine El Ghanassy*
Stabæk
1879
0.9840
0.2875
0.2355
0.1028
0.0242
0.0531
0.1261
Mohamed Elyounoussi
Molde
2275
0.9445
0.3526
0.2887
0.0509
0.0213
0.0255
0.1259
Tobias Mikkelsen*
Rosenborg
1931
0.9015
0.3444
0.2291
0.0446
0.0197
0.0630
0.1587
Moryké Fofana*
Lillestrøm
1300
0.8512
0.2796
0.2384
0.1325
0.0241
0.0176
0.0852
Ernest Asante
Stabæk
2643
0.8326
0.2662
0.2453
0.0436
0.0195
0.0683
0.1435
Samuel Adegbenro
Viking
1778
0.8245
0.3356
0.1451
0.0648
0.0212
0.0554
0.1234
Bentley
Odd
2464
0.8027
0.2561
0.1901
0.0436
0.0294
0.1292
0.0829
Ole Jørgen Halvorsen
Odd
1393
0.7688
0.2770
0.1771
0.0272
0.0271
0.1195
0.0823
0.9416
0.3206
0.2369
0.0760
0.0249
0.0643
0.1266
Top 10 average
ATTACKING MIDFIELDERS
Player
Team
Minutes
Total
Shot
Pass
Take on
Long pass
Corner
Ball carry
Iver Fossum*
Strømsgodset
2653
0.8587
0.2702
0.3297
0.0242
0.0253
0.0004
0.1227
Papa Alioune Ndiaye*
Bodø/Glimt
1286
0.7967
0.2872
0.1726
0.0670
0.0585
0.0119
0.0929
Ghayas Zahid
Vålerenga
2383
0.7618
0.2373
0.2650
0.0449
0.0048
0.0036
0.1352
Michael Barrantes*
Aalesund
1000
0.7167
0.2097
0.1888
0.0051
0.0689
0.0703
0.0817
Fredrik Nordkvelle
Odd
1891
0.6806
0.2128
0.2292
0.0198
0.0071
0.0009
0.1100
Gjermund Åsen
Tromsø
1709
0.6745
0.1602
0.1151
0.0125
0.0021
0.1190
0.0678
Bojan Zajić
Sarpsborg 08
1760
0.6574
0.2360
0.2166
0.0304
0.0140
0.0221
0.0550
Aron Elı́s Thrándarson
Aalesund
1152
0.6464
0.2980
0.1567
0.0286
0.0163
0.0095
0.0726
Daniel Fredheim Holm
Vålerenga
1974
0.6449
0.1309
0.2683
0.0439
0.0124
0.0110
0.0978
Eirik Hestad
Molde
941
0.6258
0.0661
0.2771
0.0212
0.0201
0.0846
0.0704
0.7064
0.2108
0.2219
0.0298
0.0229
0.0333
0.0906
Top 10 average
*
sold to foreign club during or some time after the 2015 season
**
on loan from a foreign club in 2015
113
Table D.5: Model 2: Top 10 players in defensive positions, Tippeligaen 2015.
Average value = 0.4608, minimum value = 0.0059
CENTRAL MIDFIELDERS
Player
Team
Minutes
Total
Shot
Pass
Cross
Long pass
Corner
Ball carry
Mike Jensen
Rosenborg
2466
1.0463
0.2562
0.3341
0.0737
0.0285
0.0929
0.1235
Etzaz Hussain*
Molde
1956
0.7513
0.1343
0.3162
0.0099
0.0458
0.0275
0.1208
Harmeet Singh
Molde
2212
0.7429
0.0985
0.3671
0.0203
0.0847
0.0128
0.0946
Giorgi Gorozia
Stabæk
1467
0.6988
0.1110
0.2833
0.0304
0.0205
0.0891
0.0612
Ole Kristian Selnæs*
Rosenborg
1951
0.6865
0.0757
0.3080
0.0319
0.0675
0.0511
0.0875
Fredrik Midtsjø
Rosenborg
2396
0.6609
0.1205
0.3048
0.0143
0.0246
-
0.0885
Jone Samuelsen
Odd
2292
0.6473
0.1435
0.2440
0.0476
0.0114
0.0024
0.0960
Malaury Martin
Lillestrøm
975
0.6427
0.1252
0.2152
0.0290
0.0442
0.1375
0.0372
Christian Grindheim
Vålerenga
2675
0.6139
0.0649
0.2832
0.0138
0.0285
0.0600
0.0567
Bismark Adjei-Boateng**
Strømsgodset 1293
0.5535
0.1999
0.2271
0.0348
0.0149
0.0007
0.0433
0.7044
0.1330
0.2883
0.0306
0.0371
0.0474
0.0809
Top 10 average
FULL BACKS
Player
Team
Minutes
Total
Shot
Pass
Cross
Corner
Throw
Ball carry
Per-Egil Flo
Molde
1998
0.7505
0.0849
0.2954
0.1053
0.1187
0.0333
0.0470
Jonas Svensson
Rosenborg
2610
0.7108
0.1126
0.3389
0.0730
-
0.0412
0.0709
Espen Ruud
Odd
2476
0.6584
0.0881
0.1954
0.1233
0.0518
0.0496
0.0471
Lars-Christopher Vilsvik
Strømsgodset 2126
0.6179
0.1047
0.2076
0.1542
0.0237
0.0371
0.0593
Martin Linnes*
Molde
2439
0.5079
0.0545
0.1931
0.0908
0.0071
0.0459
0.0430
Joachim Olsen Solberg
Mjøndalen
2486
0.4978
0.0895
0.0798
0.0517
0.1017
0.0222
0.0208
André Danielsen
Viking
2700
0.4634
0.0612
0.1425
0.0892
0.0351
0.0383
0.0523
Birger Meling
Stabæk
2263
0.4592
0.1000
0.1665
0.0279
0.0484
0.0301
0.0585
Jørgen Skjelvik
Rosenborg
1812
0.4468
0.0762
0.1873
0.0587
0.0030
0.0256
0.0620
Akeem Latifu*
Aalesund
2564
0.4454
0.0657
0.1206
0.1373
-
0.0363
0.0558
0.5558
0.0837
0.1927
0.0911
0.0389
0.0360
0.0517
Top 10 average
CENTRE BACKS
Player
Team
Minutes
Total
Shot
Pass
Aerial
Cross
Long pass
Ball carry
Johan Bjørdal
Rosenborg
1093
0.4473
0.1057
0.2274
0.0002
0.0018
0.0490
0.0803
Stefan Strandberg*
Rosenborg
1151
0.4149
0.1064
0.1797
0.0215
0.0057
0.0613
0.0883
Hólmar Örn Eyjólfsson
Rosenborg
2357
0.3758
0.0974
0.1835
0.0124
0.0005
0.0260
0.0742
Rhett Bernstein*
Mjøndalen
1234
0.3516
0.2193
0.1001
0.0650
0.0034
0.0046
0.0067
Joona Toivio
Molde
1706
0.3293
0.1184
0.1363
0.0107
0.0193
0.0259
0.0290
Jørgen Horn*
Strømsgodset 1260
0.3004
0.0737
0.1623
-0.0008
0.0125
0.0335
0.0423
Oddbjørn Lie
Aalesund
1549
0.2715
0.0495
0.1161
-0.0024
0.0381
0.0198
0.0311
Morten Sundli
Mjøndalen
2018
0.2690
0.1332
0.0907
0.0136
0.0106
0.0180
0.0205
Vegard Forren
Molde
2416
0.2614
0.0402
0.1300
0.0067
0.0123
0.0700
0.0353
Andreas Nordvik*
Sarpsborg 08
1846
0.2366
0.0830
0.0950
0.0037
0.0072
0.0630
0.0229
0.3258
0.1027
0.1421
0.0131
0.0111
0.0317
0.0430
Top 10 average
*
sold to foreign club during or some time after the 2015 season
114
**
on loan from a foreign club in 2015
Table D.6: Model 2: League table 2015 based on the model compared to the real table
Team
Value/match
Table
Real table
Diff
Rosenborg
3.9116
1
1
0
Molde
2.8190
2
6
-4
Strømsgodset
1.7488
3
2
1
Odd
1.5592
4
4
0
Stabæk
1.2503
5
3
2
Viking
0.3503
6
5
1
Vålerenga
0.1922
7
7
0
Lillestrøm
-0.9348
8
8
0
Sarpsborg 08
-0.9524
9
11
-2
Aalesund
-0.9742
10
10
0
Tromsø
-1.0293
11
13
-2
Mjøndalen
-1.1722
12
15
-3
Haugesund
-1.2649
13
12
1
Bodø/Glimt
-1.2931
14
9
5
Sandefjord
-2.0329
15
16
-1
Start
-2.1776
16
14
2
115
116
Appendix
E
Markov Game Models: Validation from
Team Evaluations
As mentioned in Sections 4.3.4 and 4.4.5 the values of the individual involvements can be
aggregated for each team each match. From this, the difference between the two opponents
in a match, in order to evaluate which team that performed the best. These values can again
be aggregated for each team over a season, from which a league table can be generated. This
can also be done after each matchday, which leads to a kind of time series for each team. The
results from such an experiment is shown in Figure E.1, where the table position of each team
is compared each match day across the two different models and with the real league table.
Figure E.1: Model 1 and Model 2: Real table position compared to estimated each matchday
The results, as can be seen from the figure, is somewhat differing amongst the teams. In or117
der to evaluate how accurate each of the models assigns positions to each team, an Augmented
Dickey Fuller (ADF) test is conducted to test the cointegration between the time series from
each model with the real league positions. In the ADF two lags are included in order to remove
any autocorrelation in the residuals. With two or more lags the test conclusions are the same,
that is if the series are cointegrated or not. In addition, both the Kendall rank correlation coefficient (Kendall’s tau) and Spearman’s rank correlation coefficent (Spearman’s rho) are calculated
for each team and each model. The results from the ADF test with two lags and the correlation
coefficients are shown in Table E.1.
Table E.1: Model 1 and Model 2: p-values from the ADF test, Kendall’s tau and Spearman’s rho in
2015
Model 1
Model 2
Team
p-value
Kendall’s tau
Spearman’s rho
p-value
Kendall’s tau
Spearman’s rho
Aalesund
0.00
0.5872
0.6788
0.00
0.4815
0.5736
Bodø/Glimt
0.02
0.7873
0.8904
0.09
-0.3181
-0.4569
Haugesund
0.01
0.5155
0.6212
0.01
0.2012
0.2346
Lillestrøm
0.12
0.2633
0.3089
0.10
0.6907
0.8283
Mjøndalen
0.01
0.8093
0.8834
0.06
-0.0270
-0.0322
Molde
0.16
0.5842
0.6962
0.03
-0.1974
-0.2206
Odd
0.06
0.7093
0.7989
0.02
-0.3636
-0.4390
Rosenborg
0.00
N/A
N/A
0.00
N/A
N/A
Sandefjord
0.00
0.8903
0.9457
0.01
0.7849
0.8614
Sarpsborg 08
0.02
0.6002
0.7494
0.25
-0.0848
-0.1149
Stabæk
0.07
0.7717
0.8171
0.00
0.4255
0.4982
Start
0.00
0.6212
0.7446
0.16
0.5587
0.6455
Strømsgodset
0.08
0.8597
0.9554
0.45
0.3536
0.4081
Tromsø
0.02
0.3029
0.3524
0.06
-0.1284
-0.1506
Viking
0.00
0.7632
0.8657
0.01
0.3542
0.3884
Vålerenga
0.14
0.7354
0.8190
0.39
0.0635
0.0912
From the table it can be seen that the test statistic for the ADF test is significant at 5%
for 10 teams with Model 1, and for 8 teams with Model 2. Furthermore, the two correlation
coefficients are positive for all teams in the first model. This indicates conformity for all the
teams regarding the real trend and the estimated development in table position. The coefficients
for Rosenborg are not applicable due to the almost absence of variations in position throughout
the season. These quite good results for Model 1 were expected, due to the heavy impact from
the scored goals.
For the second model, the two correlation coefficients are negative for six teams. This
indicates that the model estimates a somewhat opposite trend in league position than reality. A
good example of such an opposite trend can be seen through the yellow line for Bodø/Glimt
in Figure E.1. These results are a bit surprising, but very interesting since Model 2 does not
account for the outcome of each shot, and therefore is less impacted by the goals scored.
118
Appendix
F
Case Studies
In Appendix F box plots of actions performed either by Mike Jensen or Martin Ødegaard are
presented. For Mike Jensen, three plots from the 2014 season are presented, in addition to the
plot of his Model 1 values from 2015 including the shots. For Martin Ødegaard, only the plot
of his Model 1 values from 2014 is presented.
F.1
Mike Jensen
Figure F.1: Model 1: Box plot for Mike Jensen 2014 including shots
119
Figure F.2: Model 1: Box plot for Mike Jensen 2014 excluding shots
Figure F.3: Model 2: Box plot for Mike Jensen 2014
120
Figure F.4: Model 1: Box plot for Mike Jensen in 2015 including shots
F.2
Martin Ødegaard
Figure F.5: Model 1: Box plot for Martin Ødegaard 2014 including shots
121
122
Appendix
G
Results for 2016
Appendix G illustrates results from running the three models on the first 13 match days of the
2016 Tippeligaen season. The results are presented as top 20 lists, including players in all
outfield positions. A minimum of 390 minutes on the field is demanded. In the xG Model, only
players with three or more goals are considered.
G.1
Expected Goals Model
Table G.1: xG Model: Top 20 players 2016 with respect to G/xG
Name
Team
Position
Minutes Played
Goals
xG
Shots xG/S
G/xG
Milan Jevtović
Bodø/Glimt
Winger
949
6
1.79
21
0.09
3.35
Luc Kassi
Stabæk
Forward
1050
3
0.90
22
0.04
3.33
Fredrik Semb Berge
Odd
Centre Back
932
3
1.02
4
0.25
2.95
Petter Strand
Molde
Winger
940
3
1.13
20
0.06
2.66
Mathias Bringaker
Viking
Forward
565
3
1.61
17
0.09
1.87
Øyvind Storflor
Strømsgodset
Winger
859
3
1.66
15
0.11
1.81
Fredrik Nordkvelle
Odd
Attacking Midfielder
1056
4
2.23
24
0.09
1.79
Bassel Jradi
Strømsgodset
Attacking Midfielder
1049
3
1.71
26
0.07
1.75
Fredrik Gulbrandsen
Molde
Forward
515
4
2.29
17
0.13
1.75
Bentley
Odd
Winger
994
3
1.77
20
0.09
1.69
Torbjørn Agdestein
Haugesund
Forward
944
6
3.55
24
0.15
1.69
Malaury Martin
Lillestrøm
Central Midfielder
1097
4
2.43
34
0.07
1.65
Fred Friday
Lillestrøm
Forward
1080
8
4.87
50
0.10
1.64
Steffen Ernemann
Sarpsborg 08
Central Midfielder
565
3
1.91
9
0.21
1.57
Marcus Pedersen
Strømsgodset Forward
670
5
3.29
24
0.14
1.52
Thomas Lehne Olsen
Tromsø
Forward
712
3
2.01
16
0.13
1.49
Matthı́as Vilhjálmsson
Rosenborg
Forward
726
3
2.02
20
0.10
1.49
Tokmac Nguen
Strømsgodset
Attacking Midfielder
912
3
2.09
24
0.09
1.43
Kristoffer Tokstad
Sarpsborg 08
Winger
1013
3
2.32
21
0.11
1.29
Deshorn Dwayne Brown
Vålerenga
Forward
991
6
4.95
45
0.11
1.21
123
G.2
Markov Game Model 1
Table G.2: Model 1: Top 20 players in all positions 2016.
Average tota value = 0.0752, minimum total value = -0.4143
Name
Team
Position
Minutes Played Total value
Milan Jevtović
Bodø/Glimt
Winger
949
0.4873
Fredrik Gulbrandsen
Molde
Forward
515
0.4612
Haris Hajradinović
Haugesund
Winger
670
0.4176
Steffen Ernemann
Sarspborg 08
Central midfielder
565
0.4174
Fredrik Midtsjø
Rosenborg
Central midfielder
1006
0.4167
Fredrik Semb Berge
Odd
Centre back
932
0.3566
Fred Friday
Lillestrøm
Forward
1080
0.3492
Torbjørn Agdestein
Haugesund
Forward
944
0.3266
Simen Kind Mikalsen Lillestrøm
Full back
900
0.3172
Moussa Nije
Stabæk
Winger
547
0.3097
Mattias Moström
Molde
Winger
421
0.2862
Alexander Gersbach
Rosenborg
Full back
440
0.2807
Petter Strand
Molde
Winger
940
0.2685
Malaury Martin
Lillestrøm
Central midfielder
1097
0.2682
Øyvind Storflor
Strømsgodset Winger
859
0.2659
Per-Egil Flo
Molde
Full back
990
0.2657
Filip Kiss
Haugesund
Central midfielder
1010
0.2616
Mathias Bringaker
Viking
Forward
565
0.2490
Mos
Aalesund
Forward
656
0.2462
Fredrik Nordkvelle
Odd
Attacking midfielder
1056
0.2408
124
G.3
Markov Game Model 2
Table G.3: Model 2: Top 20 players in all positions 2016.
Average total value = 0.4541, minimum total value = 0.0356
Name
Team
Position
Minutes Played
Total value
Pål André Helland
Rosenborg
Winger
505
1.1727
Ghayas Zahid
Vålerenga
Attacking midfielder
949
0.9958
Mike Jensen
Rosenborg
Central midfielder
1080
0.9790
Mohamed Elyounoussi
Molde
Winger
971
0.9497
Mahatma Otoo
Sogndal
Forward
624
0.9487
Gilbert Koomson
Sogndal
Winger
948
0.9293
Fred Friday
Lillestrøm
Forward
1080
0.8992
Samuel Adegbenro
Viking
Winger
1079
0.8878
Yann-Erik de Lanlay
Rosenborg
Winger
967
0.8802
Olivier Occéan
Odd
Forward
1080
0.8486
Patrick Mortensen
Sarpsborg 08
Forward
860
0.8295
Malaury Martin
Lillestrøm
Central midfielder
1097
0.8152
Øyvind Storflor
Strømsgodset Winger
859
0.8076
Anders Konradsen
Rosenborg
Central midfielder
1168
0.7993
Ole Jørgen Halvorsen
Odd
Winger
418
0.7957
Espen Ruud
Odd
Full back
1080
0.7938
Fredrik Midtsjø
Rosenborg
Central midfielder
1006
0.7918
Sofien Moussa
Tromsø
Forward
617
0.7888
Christian Gytkjær
Rosenborg
Forward
840
0.7834
Attacking midfielder
738
0.7508
Daniel Fredheim Holm Vålerenga
125
126

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